CRE 2 - Triangles | Geometry, Mensuration & Trigonometry

Answer: Option B

Explanation :

In a right triangle, one angle is 90° and the sum of the other two angles is 90°.

These acute angles are divided in the ratio of 1 : 5.

∴ We get x + 4x = 90° 

∴ x = 18°

Hence, we get the two acute angles as 18° and 72°. 

∴ The smallest angle of the triangle is 18°.

Hence, option (b).

Answer: Option C

Explanation :

Altitude of an equilateral triangle is given by √3/2 × a.

where a is the side of an equilateral triangle.

∵ a = 4 cm

∴ Altitude = √3/2 × 4 = 2√3 cm.

Hence, option (c).

Answer: Option C

Explanation :

In a triangle the sum of any two sides should be more than the third side and the difference of any two sides should be less than the third side. 

Only option (c) satisfies both the conditions.

Hence, the length of the third side can be 6 units.

Hence, option (c).

Answer: Option B

Explanation :

s = (a + b + c)/2

where s is the semi-perimeter and a, b and c are the sides of the triangle.

s = (6 + 8 + 10)/2 = 12 cm

Area = $\sqrt{\mathrm{s}\left(\mathrm{s}-\mathrm{a}\right)\left(\mathrm{s}-\mathrm{b}\right)\left(\mathrm{s}-\mathrm{c}\right)}$

        = $\sqrt{12\left(12-6\right)\left(12-8\right)\left(12-10\right)}=\sqrt{12\times 6\times 4\times 2}$

        = 24

Alternately,

6, 8 and 10 are pythagorean triplets i.e. 10² = 8² + 6²

⇒ The given triangle is a right triangle

⇒ Area = ½ × 6 × 8 = 24

Hence, option (b).

Answer: Option C

Explanation :

Sum of the interior angles of a triangle is 180°. 

These angles are in the ratio 3 : 4 : 5.

∴ 3x + 4x + 5x = 180°

∴ x = 15°

∴ We get the three angles as 45°, 60° and 75° respectively. 

The greatest external angle should be supplementary to the smallest internal  angle which measures 45°.

∴ Measure of the greatest external angle = 180 − 45° = 135°

Hence, option (c).

Answer: Option A

Explanation :

BC is the base of ∆ABC and BD is base of ∆ABD. 

If we drop a perpendicular from A on the line BC, this line is the height for both, ∆ABC and  ∆ABD

∴ The ratio of the areas of the two triangles is equal to the ratio of their bases.

∴ A(∆ABD) = 1/5 × A(∆ABC)

∴ A(∆ABD) = 1/5 × 100 = 20 sq. units

Hence, option (a).

Answer: Option C

Explanation :

In a triangle with sides a, b and c, sum of any two sides is greater than the third side, i.e. a + b > c.

∴ For any three sides given, a triangle can be constructed only if sum of any two sides is greater than the third side.

It is given that, a + b + c = 14

Also, it is given that a, b, c are integers.

Case 1: The largest side is 6.
The possible combination of sides is (6, 4, 4), (6, 5, 3) and (6, 6, 2)

Case 2: The largest side is 5.
The possible combination of sides is (5, 5, 4)

∴ The possible sets are (4, 4, 6), (5, 5, 4), (6, 5, 3) and (6, 6, 2).

∴ Only 4 such triangles are possible.

Hence, option (c).

Answer: 90

Explanation :

Since two sides of the triangle are in the ratio 1 : 1, these two sides of the triangle should be equal. 

∴ The given triangle is isosceles in nature. 

One of the sides is √2 times the other two sides

∴ By the 45°-45°-90° theorem, the given triangle is an Isosceles Right angled triangle.

∴ The largest angle is 90°

Hence, 90.

Answer: Option E

Explanation :

Given external angle C = 150°

⇒ internal angle C = 180° - 150° = 30°

∴ m ∠ACB = 30°

AC = BC = 10 cm

Two sides and the angle included by them are known.

∴ Area of the triangle = 1/2 × (AC) × (BC) × sin θ

∴ Area of  ∆ABC = 1/2 × (10) × (10) × sin 30° = 25 cm². [sin 30° = 1/2]

Hence, option (e).

Answer: Option A

Explanation :

Let the interior angle be 2x and exterior angle be 3x.

The interior and the corresponding exterior angle of a triangle are supplementary.

⇒ 2x + 3x = 180°
⇒ x = 36°

Now, exterior angle is same as the sum of two opposite interior angles.

∴ Sum of the other 2 interior angles of the triangle = 3 × 36° = 108°

Hence, option (a).

Answer: Option A

Explanation :

The three angles of the triangle are in the ratio 1 : 2 : 3.

Hence, the given triangle can only be a 30°-60°-90° triangle.

Here, the side opposite the largest angle is the hypotenuse and the side opposite the smallest angle is the side opposite the angle with measure 30°.

Hence, by the 30°-60°-90° theorem

Side opposite to 30° angle is ½ of hypotenuse

∴ Ratio of the length of the largest side to the length of the smallest side is the ratio of the hypotenuse to the side opposite the angle with measure 30° 

∴ (Side opposite to 30°)/Hypotenuse = 1/2

Hence, option (a).

Answer: Option D

Explanation :

We can draw the following diagram from the information given below.

​​​​​​​

In △ABD: AB² = BD² + AD²
⇒​​​​​​​ AD² = 169 - x²   ...(1)

In △ACD: AC² = CD² + AD²
⇒​​​​​​​ AD² = 225 - (14 - x)²   ...(2)

(1) = (2)
⇒​​ 169 - x² = 225 - (14 - x)²
⇒​​ 28x = 140
⇒​​ x = 5

⇒​​ AD² = 169 - 25 = 144 [from (1)]
⇒​​ AD = 12​​​​​​​

Hence, option (d).

Answer: Option D

Explanation :

Let us draw a line joining points C and A and extend it till E.

​​​​​​​

In ∆ABC, ∠EAB is the exterior angle, hence
∠EAB = ∠ABC + ∠BCA   ...(1)

In ∆ADC, ∠EAD is the exterior angle, hence
∠EAD = ∠ADC + ∠DCA   ...(2)

(1) + (2)

∠EAB + ∠EAD = ∠ABC + ∠BCA + ∠ADC + ∠DCA
⇒​​​​​​​ ∠BAD = ∠ABC + ∠ADC + ∠BCD
⇒​​​​​​​ ∠BAD = 30° + 10° + 20° = 60°

Hence, option (d).

Answer: Option B

Explanation :

∆BDE and ∆CDE lie between same 2 parallel lines, hence the ratio of their areas is same as the ratio of their bases.

⇒ $\frac{\mathrm{area}(∆\mathrm{BDE})}{\mathrm{area}(∆\mathrm{CDE})}$ = $\frac{3}{4}$   ...(1)

Similarly, $\frac{\mathrm{area}(∆\mathrm{ADB})}{\mathrm{area}(∆\mathrm{BDE})}$ = $\frac{3}{2}$   ...(2)

(1) × (2)

⇒ $\frac{\mathrm{area}(∆\mathrm{ADB})}{\mathrm{area}(∆\mathrm{CDE})}$ = $\frac{9}{8}$

Hence, option (b).

Answer: Option C

Explanation :

AD is the angle bisector of ∠A = 100°.
∴ ∠BAD = ∠CAD = 50°

We know side opposite to larger angle is larger than side opposite to smaller angle.

In triangle ABD,
∠ABD > ∠BAD
⇒ AD > BD   ...(1)

In triangle ADC,
∠DAC > ∠ACD
⇒ DC > AD   ...(2)

From (1) and (2) 
⇒ DC > AD > BD

Hence, option (c).

Answer: Option B

Explanation :

In a triangle the sum of three altitudes of a triangle is less than the sum of sides.

Hence, option (b).

Answer: Option B

Explanation :

Since AB = AC, it is an isosceles triangle. The perpendicular drawn from A to BC will bisect side BC.

​​​​​​​

In right triangle ADB,
AB = 10 and BD = 6, hence AD = 8 [6, 8 and 10 is a pythagorean triplet]

∴ Area of triangle = ½ × BC × AD = ½ × 12 × 8 = 48.

Hence, option (b).

Answer: Option A

Explanation :

For same perimeter area of an equilateral triangle is maximum.

Hence, option (a).

Answer: Option C

Explanation :

For same area, perimeter of an equilateral triangle is minimum.

Hence, option (c).

Answer: Option D

Explanation :

We can apply the cosine rule to find the third side.

AC² = AB² + BC² - 2 × AB × BC × CosB

Here, AB = 6, BC = 4 and ∠B = 60°

⇒ AC² = 6² + 4² - 2 × 6 × 4 × Cos60°

⇒ AC² = 36 + 16 - 2 × 6 × 4 × 1/2 [Cos60°  = 1/2]

⇒ AC² = 36 + 16 - 6 × 4

⇒ AC² = 28

⇒ AC = 2√7

Hence, option (c).

Answer: Option B

Explanation :

5, 12 and 13 form a pythagorean tiplet, hence the given triangle is a right triangle with AC as the hypotenuse

In a right triangle, the length of altitude drawn on hypotenuse is $\sqrt{\frac{\mathrm{product} \mathrm{of} \mathrm{smaller} \mathrm{sides}}{\mathrm{hypotenuse}}}$.

∴ altitude = $\sqrt{\frac{5\times 12}{13}}$ = $\sqrt{\frac{60}{13}}$

Hence, option (b).