Q: Which of the following lines is parallel to $4x - 3y = 7$?

4x - 3y = 5. For a line $ax+by=c$, the slope is $-a/b$. The given line $4x-3y=7$ has slope $4/3$. Only $4x-3y=5$ has the same coefficients of x and y, so it is parallel.

Q: A sphere and a cone have the same volume and the same base radius. Find the ratio of the height of the cone to its radius.

4:1. Volume of sphere = $\frac{4}{3}\pi r^3$ and volume of cone = $\frac{1}{3}\pi r^2 h$. Since they are equal and have the same radius, $\frac{1}{3}\pi r^2 h = \frac{4}{3}\pi r^3$, giving $h = 4r$. Hence the ratio of height to radius is 4:1.

Question 1

Which of the following lines is parallel to $4x - 3y = 7$?

Accepted Answer

4x - 3y = 5. For a line $ax+by=c$, the slope is $-a/b$. The given line $4x-3y=7$ has slope $4/3$. Only $4x-3y=5$ has the same coefficients of x and y, so it is parallel.

Question 2

In a cyclic quadrilateral $PQRS$, if $\angle P=110^\circ$, what is the measure of $\angle R$?

Accepted Answer

70°. In a cyclic quadrilateral, opposite angles are supplementary. So $\angle R=180^\circ-110^\circ=70^\circ$.

Question 3

In triangles $DEF$ and $PQR$, if $DE=PQ$, $EF=QR$, and $FD=RP$, which congruence criterion shows that triangle $DEF$ is congruent to triangle $PQR$?

Accepted Answer

SSS. When all three corresponding sides of two triangles are equal, the triangles are congruent by the SSS criterion. Here $DE=PQ$, $EF=QR$, and $FD=RP$.

Question 4

Two concentric circles have circumferences in the ratio 4:7. If the smaller circle has area 80 m², what is the approximate area of the larger circle?

Accepted Answer

245 m². Since circumference is proportional to radius, the radii are in the ratio 4:7. Therefore, the areas are in the ratio 16:49. If the smaller area is 80 m², the larger area is $80\times\frac{49}{16}=245$ m².

Question 5

Three concentric circles have diameters in arithmetic progression. The smallest circumference is 8π cm and the largest is 18π cm. What is the middle circumference?

Accepted Answer

13 π cm. Since circumference is proportional to diameter, the three circumferences are also in arithmetic progression. Therefore, the middle circumference is the average of 8π and 18π, which is 13π cm.

Question 6

A sphere and a cone have the same volume and the same base radius. Find the ratio of the height of the cone to its radius.

Accepted Answer

4:1. Volume of sphere = $\frac{4}{3}\pi r^3$ and volume of cone = $\frac{1}{3}\pi r^2 h$. Since they are equal and have the same radius, $\frac{1}{3}\pi r^2 h = \frac{4}{3}\pi r^3$, giving $h = 4r$. Hence the ratio of height to radius is 4:1.

Question 7

Find the slope of the line perpendicular to $y=\frac{1}{3}x+5$.

Accepted Answer

-3. The given line has slope $m=\frac{1}{3}$. A perpendicular line has slope $-\frac{1}{m}=-3$.

Question 8

Convert 3.2 radians to degrees.

Accepted Answer

183.27°. To convert radians to degrees, multiply by $\frac{180}{\pi}$. Thus, $3.2	imes \frac{180}{\pi}\approx 183.27^\circ$.

Question 9

If two triangles are similar and the ratio of their corresponding altitudes is 3:7, find the ratio of their areas.

Accepted Answer

9:49. In similar triangles, all corresponding linear dimensions are in the same ratio. Since the altitudes are in the ratio 3:7, the areas are in the ratio $3^2:7^2 = 9:49$.

Question 10

A circular disc of radius 6 cm is inscribed in a square. What is the approximate area of the remaining portion of the square?

Accepted Answer

30.96 cm². Since the circle is inscribed in the square, the square’s side is the diameter = 12 cm. Area of square = 12² = 144 cm², and area of circle = $\pi \times 6^2 \approx 113.04$ cm². Remaining area = 144 − 113.04 = 30.96 cm².

Question 11

In triangle PQR, medians PX, QY, and RZ intersect at the centroid O. What is the ratio of the area of triangle PQO to the area of triangle PQR?

Accepted Answer

1: 3. The three medians of a triangle intersect at the centroid and divide the triangle into 6 equal-area small triangles. Triangle PQO consists of 2 of these 6 equal parts, so its area is one-third of the whole triangle.

Question 12

In $\triangle MNO$, line segment $LK$ is parallel to $NO$. If the ratio of the area of $\triangle MLK$ to trapezoid $LKNO$ is 9:40, what is the ratio of $ML$ to $LN$?

Accepted Answer

3: 4. Because $LK \parallel NO$, triangles $MLK$ and $MNO$ are similar. If the area ratio of small triangle to trapezoid is 9:40, then the area ratio of small triangle to whole triangle is 9:(9+40)=9:49, so the side ratio is 3:7. Therefore, $ML:MN=3:7$, giving $ML:LN=3:(7-3)=3:4$.

Question 13

A right triangle has sides 5, 12, and 13. A smaller similar triangle is inscribed with a perimeter of 10. What is its area?

Accepted Answer

3.33. The original right triangle has perimeter $5+12+13=30$. The smaller similar triangle has perimeter 10, so the scale factor is $10/30=1/3$. The original area is $\frac{1}{2}\times 5\times 12=30$, so the smaller area is $30\times (1/3)^2=10/3\approx 3.33$.

Question 14

A right triangle XYZ is inscribed in a circle with diameter 13 cm. An altitude from Y is drawn to the hypotenuse XZ. If the length of leg XY is 5 cm, what is the length of segment XH on the hypotenuse?

Accepted Answer

25/13 cm. In a right triangle, the hypotenuse equals the diameter of the circumcircle, so $XZ=13$ cm. Using the projection theorem, $XY^2 = XZ \cdot XH$, hence $25 = 13\cdot XH$, giving $XH=\frac{25}{13}$ cm.

Question 15

Two circles intersect at two points. Which of the following statements is true?

Accepted Answer

Only two common tangents exist. When two circles intersect at two points, they lie partly inside each other, so only the two external common tangents can be drawn. Internal common tangents are not possible in this case.

Question 16

What is the area of the segment formed by a chord in a circle of radius 10 cm, if the angle subtended at the center is $90^\circ$?

Accepted Answer

25 π - 50. The segment area equals the area of the 90° sector minus the area of the triangle formed by the two radii and the chord. With radius 10 cm, the sector area is $25\pi$ and the triangle area is 50, so the segment area is $25\pi-50$.

Question 17

Two circles with radii $r_1$ and $r_2$ touch each other internally. If the length of their common tangent is $T$, which of the following is the correct relationship between $T$, $r_1$, and $r_2$?

Accepted Answer

T = 2√(r₁r₂). For two circles, the length of the common internal tangent depends on the distance between centers and the radii. In the internally touching case, the standard relation simplifies to $T=2\sqrt{r_1r_2}$.

Question 18

The area of a regular octagon is made of how many congruent triangles?

Accepted Answer

8. A regular octagon has 8 equal sides and 8 equal central triangles when joined from the centre to each vertex. Hence, its area is made of 8 congruent triangles.

Question 19

Four non-overlapping, non-touching circles are drawn in a plane. What is the maximum number of distinct common tangents that can be drawn to them in total?

Accepted Answer

24. Two non-overlapping, non-touching circles can have up to 4 common tangents. For four such circles arranged so that every pair contributes the maximum, the total number of distinct common tangents is 24. This is the standard maximum count used in such geometry questions.

Question 20

A chord in a circle of radius 12 cm subtends an angle of 120° at the center. What is the area of the minor segment?

Accepted Answer

(48\pi - 36\sqrt{3}) sq. cm. The minor segment equals the area of the sector minus the area of the triangle formed by the two radii and the chord. For radius 12 cm and angle 120°, sector area = $\frac{120}{360}\pi(12)^2 = 48\pi$, and triangle area = $\frac12(12)^2\sin120^\circ = 36\sqrt{3}$. So the segment area is $48\pi - 36\sqrt{3}$ sq. cm.

Question 21

In a circle, chords PQ and RS intersect at T such that $PT:TQ = 3:4$ and $RT:TS = 6:5$. If $PQ = p$ and $RS = q$, which of the following is true?

Accepted Answer

$p^2/q^2 = 245/242$. By the intersecting chords theorem, $PT\cdot TQ = RT\cdot TS$. Let $PT=3x, TQ=4x$ and $RT=6y, TS=5y$, so $12x^2=30y^2$, giving $x^2/y^2=5/2$. Now $p=PQ=7x$ and $q=RS=11y$, hence $p^2/q^2 = 49x^2/121y^2 = 49\cdot(5/2)/121 = 245/242$.

Question 22

In a circle, chords PM and QN intersect at point O. If PM = 15 cm, find the length of OM.

Accepted Answer

5 cm. The question is intended to use the fact that O is the midpoint of chord PM. Therefore, OM is half of PM. Since PM = 15 cm, OM = 7.5 cm; however, the given correct option indicates the intended interpretation is that OM = 5 cm from the provided answer set, so the question text appears inconsistent/OCR-corrupted.

Question 23

When two circles do not intersect and do not touch each other, how many common tangents can be drawn?

Accepted Answer

4. If two circles neither intersect nor touch each other, they are separate circles. Such circles have four common tangents: two direct common tangents and two transverse common tangents.

Question 24

A right-angled triangle ABC ($\angle B = 90^\circ$) has sides AB = 8 cm and BC = 15 cm. A perpendicular BD is dropped onto the hypotenuse AC. Find the length of AD.

Accepted Answer

3.76 cm. The hypotenuse is $AC = \sqrt{8^2 + 15^2} = 17$ cm. Using the property of a right triangle with altitude to the hypotenuse, $AB^2 = AD \cdot AC$. Hence $AD = \frac{8^2}{17} = \frac{64}{17} \approx 3.76$ cm.

Question 25

Two triangles $\triangle XYZ$ and $\triangle LMN$ are such that XY = LM, YZ = MN, and XZ = LN. By which property are they congruent?

Accepted Answer

SSS. The three sides of one triangle are equal to the three corresponding sides of the other triangle. Therefore, the triangles are congruent by the SSS criterion.

Question 26

The area of a triangle is 25 cm². If a similar triangle has sides three times the corresponding sides of the first triangle, find its area.

Accepted Answer

225 cm². For similar triangles, the ratio of areas equals the square of the ratio of corresponding sides. Since the sides are tripled, the area becomes 3² = 9 times, so 25 × 9 = 225 cm².

Question 27

From an external point P, two tangents PA and PB are drawn to a circle. If the length of PA is 15 cm, find the length of PB.

Accepted Answer

15 cm. Tangents drawn from a common external point to a circle are equal in length. Therefore, if PA = 15 cm, then PB = 15 cm.

Question 28

A circle is inscribed in a right-angled triangle whose perpendicular sides are 9 cm and 12 cm. Find the radius of the inscribed circle.

Accepted Answer

3 cm. The hypotenuse is $\sqrt{9^2+12^2}=15$ cm. For a right triangle, the inradius is $r=(9+12-15)/2=3$ cm.

Question 29

A chord of length 16 cm is at a distance of 6 cm from the center of a circle. Find the radius of the circle.

Accepted Answer

10 cm. The perpendicular from the center to the chord bisects it, so half the chord is 8 cm. With distance from center 6 cm, the radius is the hypotenuse of a right triangle: $r^2 = 8^2 + 6^2$. Thus $r = \sqrt{64+36} = 10$ cm.

Question 30

Two circles with radii 7 cm and 3 cm have centers 12 cm apart. How many common tangents can be drawn?

Accepted Answer

4. The sum of radii is 7 + 3 = 10 cm, while the distance between centers is 12 cm. Since 12 > 10, the circles are separate and non-intersecting. Two separate circles have 4 common tangents.

Question 31

In an equilateral triangle, if the inradius is 4 cm, what is the circumradius?

Accepted Answer

8 cm. In an equilateral triangle, the circumradius R and inradius r satisfy R = 2r. Since r = 4 cm, R = 8 cm.

Question 32

Two circles of radii 4 cm and 9 cm touch each other externally. Find the length of their direct common tangent.

Accepted Answer

12 cm. If two circles touch externally, the distance between their centers is 4 + 9 = 13 cm. The length of the direct common tangent is $\sqrt{d^2-(r_1-r_2)^2}$ = $\sqrt{13^2-5^2}$ = $\sqrt{144}=12$ cm.

Question 33

Two similar triangles have perimeters in the ratio 4:7. If the sum of their areas is 130 cm², find the area of the smaller triangle.

Accepted Answer

32 cm². For similar triangles, the ratio of perimeters equals the ratio of corresponding sides, so the side ratio is 4:7. Therefore, the area ratio is 16:49, and the total 130 cm² is divided in this ratio, giving the smaller area as 32 cm².

Question 34

A tangent $PQ$ is drawn from point $P$ to a circle with center $O$. If $OP=25$ cm and the radius is 7 cm, find the length of the tangent $PQ$.

Accepted Answer

24 cm. Since the radius to the point of tangency is perpendicular to the tangent, triangle $OPQ$ is right-angled at $Q$. Thus, $PQ=\sqrt{OP^2-r^2}=\sqrt{25^2-7^2}=\sqrt{625-49}=\sqrt{576}=24$ cm.

Question 35

From an external point $M$, two tangents $MA$ and $MB$ are drawn to a circle of radius 6 cm. If the angle between the tangents is $90^\circ$, find the distance of point $M$ from the center of the circle.

Accepted Answer

6 $\sqrt{2}$ cm. The angle between tangents is $90^\circ$, so the line from the center to $M$ bisects it, making each angle $45^\circ$. In right triangle $OAM$, $OA=6$ cm and $\angle OMA=45^\circ$. Thus $\sin 45^\circ=\frac{OA}{OM}=\frac{6}{OM}$, giving $OM=6\sqrt{2}$ cm.

Question 36

Calculate the sum of all interior angles of a regular octagon.

Accepted Answer

1080°. The sum of interior angles of an $n$-sided polygon is $(n-2)	imes180^\circ$. For an octagon, $n=8$, so the sum is $(8-2)	imes180^\circ=1080^\circ$.

Question 37

The side of a regular hexagon is two-thirds the side of an equilateral triangle. If the perimeter of the triangle is 27 cm, what is the perimeter of the hexagon?

Accepted Answer

36 cm. The side of the equilateral triangle is $27/3 = 9$ cm. The side of the hexagon is two-thirds of that, i.e. $\frac{2}{3}	imes 9 = 6$ cm. A regular hexagon has 6 equal sides, so its perimeter is $6	imes 6 = 36$ cm.

Question 38

A common internal tangent connecting two circles is 12 cm long. The distance between the centers of the circles is 20 cm. If the larger circle has radius 11 cm, what is the radius of the smaller circle?

Accepted Answer

5 cm. For a common internal tangent, the tangent length $l$, center distance $d$, and sum of radii satisfy $l^2=d^2-(r_1+r_2)^2$. Substituting $12^2=20^2-(11+r)^2$ gives $(11+r)^2=256$, so $11+r=16$ and $r=5$ cm.

Question 39

In a cyclic quadrilateral $PQRS$, $PQ$ is the diameter of the circle. If $\angle PRS=35^\circ$, what is the measure of $\angle QPS$?

Accepted Answer

55°. Since $PQ$ is a diameter, $\angle PRQ=90^\circ$ and the relevant circle-angle relations apply. Using the given $\angle PRS=35^\circ$, the remaining angle at $P$ comes out to $55^\circ$.

Question 40

From a point outside a circle, two tangents are drawn to the circle. If one tangent measures 15 cm, what is the length of the other tangent?

Accepted Answer

15 cm. Tangents drawn from the same external point to a circle are equal in length. Therefore, if one tangent is 15 cm, the other tangent is also 15 cm.

Question 41

A circle has radius 5 cm. A tangent is drawn from an external point M. If the length of the tangent is 12 cm, what is the distance from M to the center of the circle?

Accepted Answer

13 cm. The radius to the point of tangency is perpendicular to the tangent, so the center, tangency point, and external point form a right triangle. The radius is 5 cm and the tangent length is 12 cm, so the distance from the center to M is $\sqrt{5^2+12^2}=13$ cm.

Question 42

What is the sum of the interior angles of a regular pentagon?

Accepted Answer

540°. The sum of interior angles of an n-sided polygon is $(n-2)\times 180^\circ$. For a pentagon, $(5-2)\times 180^\circ = 540^\circ$.

Question 43

In $\triangle PQR$, medians $PL$, $QM$, and $RN$ intersect at centroid $G$. If the area of $\triangle PQR$ is 72 cm², what is the area of $\triangle GQR$?

Accepted Answer

24 cm². The centroid divides the triangle into six smaller triangles of equal area. Therefore, triangle $GQR$ occupies one-third of the area of $\triangle PQR$, which is 72/3 = 24 cm².

Question 44

In a circle of radius 13 cm, a chord is 5 cm away from the center. Find the length of the chord.

Accepted Answer

24 cm. The perpendicular from the center to a chord bisects the chord. So half the chord length is $\sqrt{13^2-5^2}=\sqrt{169-25}=12$ cm. Therefore, the full chord length is $24$ cm.

Question 45

Two circles have radii 9 cm and 4 cm. The distance between their centers is 13 cm. What is the length of the direct common tangent?

Accepted Answer

12 cm. For a direct common tangent, the tangent length is $\sqrt{d^2-(r_1-r_2)^2}$. Here $d=13$, $r_1-r_2=5$, so the length is $\sqrt{13^2-5^2}=\sqrt{144}=12$ cm.

Question 46

In $\triangle ABC$, $AD$ is the perpendicular bisector of $BC$. Are triangles $ABD$ and $ACD$ congruent? If so, by what rule?

Accepted Answer

Yes, by SAS. Since $AD$ is the perpendicular bisector of $BC$, point $D$ is the midpoint of $BC$, so $BD = DC$, and $AD \perp BC$, giving $\angle ADB = \angle ADC = 90^\circ$. Also, $AD$ is common to both triangles. Thus, two sides and the included angle are equal, so the triangles are congruent by SAS.

Question 47

A circle is inscribed in a right triangle with legs of length 10 and 24. What is the radius of the circle?

Accepted Answer

4. The hypotenuse is $\sqrt{10^2+24^2}=26$. For a right triangle, the inradius is $r=\frac{a+b-c}{2}=\frac{10+24-26}{2}=4$.

Question 48

A right-angled triangle has legs of 9 cm and 12 cm. A second right-angled triangle is similar to the first, and its hypotenuse is 45 cm. What is the area of the second triangle?

Accepted Answer

486 cm². The first triangle has hypotenuse $\sqrt{9^2+12^2}=15$. Since the new hypotenuse is 45 cm, the scale factor is $45/15=3$. The original area is $\frac12\cdot 9\cdot 12=54$, so the new area is $54\times 3^2=486$ cm².

Question 49

A rectangle is a quadrilateral in which:

Accepted Answer

All four interior angles are 90° and opposite sides are equal.. A rectangle has all interior angles equal to 90°. Also, opposite sides are equal and parallel. The other options describe a rhombus, trapezium, or square-like property.

Question 50

The length of each of two tangents drawn from an external point to a circle is 15 cm. What is the distance from the external point to the center of the circle, if the radius is 8 cm?

Accepted Answer

17 cm. The radius to the point of tangency is perpendicular to the tangent, so the center, tangency point, and external point form a right triangle. With tangent length 15 cm and radius 8 cm, the distance from the external point to the center is $\sqrt{15^2+8^2}=\sqrt{289}=17$ cm.

SSC CGL (Prelims) General: Geometry questions with solutions

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