Mathematics · Quantitative Aptitude

Triangle Properties

243 Questions

Triangle properties encompass the rules governing the sides, angles, and area of different types of triangles. Key areas include the Pythagorean theorem, similar triangles, and centroid calculations. These concepts form a foundational part of geometry in various competitive examinations.

Similar trianglesArea and perimeterPythagorean theoremTriangle inequalityEquilateral properties

Triangle Properties Questions

Multiple choice maths congruence introduction to shapes similarity of triangles introduction to similar triangles

The perimeter of two similar triangles $\triangle ABC$ and $\triangle DEF$ are $36$ cm and $24$ cm respectively. If $DE=10 $ cm, then $AB$ is :

  1. $12$ cm
  2. $20$ cm
  3. $15$ cm
  4. $18$ cm
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given that triangles $ABC$ and $DEF$ are similar.

Also given, $DE=10$ cm and perimeters of triangles $ABC$ and $DEF$ are $36$ cm and $24$ cm.
So, the corresponding sides of the two triangles is equal to the ratio of their perimeters.

Hence, $\dfrac {\text{perimeter of} \ ABC}{ \text{perimeter of } \ DEF}$ $=\dfrac {AB}{DE}$
Therefore, $\dfrac {36}{24}=\dfrac {AB}{10}$ 
$\Rightarrow AB=\dfrac {36\times 10}{24}$
$\Rightarrow AB=15$ cm

Multiple choice maths congruence introduction to shapes similarity of triangles introduction to similar triangles

The sides of a triangle are $5$ cm, $6$ cm and $7$ cm. One more triangle is formed by joining the midpoints of the sides. The perimeter of the second triangle is:

  1. $18$ cm
  2. $12$ cm
  3. $9$ cm
  4. $6$ cm
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let the $\triangle ABC $ have sides $AB = 5$cm,

$BC = 6$ cm and $AC = 7$cm.
Let the midpoints of the sides AB and AC be points D and E respectively.
$\therefore \dfrac {AD}{DB} = \dfrac {AE}{EC}$         ...By B.P.T

$\therefore \dfrac {AD+DB}{DB} = \dfrac {AE+EC}{EC}$   ....By Componendo

$\therefore \dfrac {AB}{DB} = \dfrac {AC}{EC}$      ......(1)

Also, $\angle BAC \cong \angle DAE$    ....(2)

$\therefore \triangle ABC \sim \triangle ADE$      ....SAS test of similarity

$\therefore \dfrac {AB}{AD} = \dfrac {BC}{DE} = \dfrac {AC}{AE}$       ....C.S.S.T

But $\dfrac {AB}{AD} = \dfrac {AB}{\frac 12 AB} = \dfrac 12$

$\therefore \dfrac {BC}{DE} = \dfrac 12$


Perimeter $(\triangle ADE) = AD + DE + AE$ 
$ = \dfrac 12 AB + \dfrac 12 BC + \dfrac 12 AC$

$= \dfrac 12 \left(AB + BC + AC \right)$

$ = \dfrac 12 \times 18 = 9$ cm.

So, option C is correct.

Multiple choice maths geometry similarity and right angled triangle angle theorems for a right angled triangle apollonius's theorem

Let $\Delta _1$ denotes the area of the triangle formed by the vertices $(a^3m^3 _1, am _1), (a^3m^3 _2am _2), (a^3m^3 _3, am _3)$ and $\Delta _2$ denotes the area of the triangle formed by the vertices $(2am _1m _2, a^2(m^2 _1+m^2 _2))$, $(2am _2m _3, a^2(m^2 _2+m^2 _3))$ and $(2am _3m _1, a^2(m^2 _3+m^2 _1))$. Then $\dfrac{\Delta _1}{\Delta _2}(a > 0)$ equals?

  1. $\dfrac{a}{2}$
  2. $2a$
  3. $\dfrac{a^3}{8}$
  4. $8a^3$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Calculating the area of the triangles using the determinant formula for coordinates and simplifying the ratio yields a/2.

Multiple choice maths geometry similarity and right angled triangle angle theorems for a right angled triangle apollonius's theorem

Find the perimeter of an isosceles right triangle with each of its congruent as 7cm.

  1. $7\sqrt 2$ cm
  2. $14$ cm
  3. $(2+ \sqrt 2)$ cm
  4. $7(2+ \sqrt 2)$ cm
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let the other side of triangle is x cm

Then in isosceles right angle triangle two congruent sides are 7 cm
$x^{2}=(7)^{2}+(7)^{2}$
$\Rightarrow x^{2}=49+49$
$\Rightarrow x^{2}=198$
$\Rightarrow x=7\sqrt{2}$
Then perimeter of right angle isosceles triangle =$7+7+7\sqrt{2}=14+7\sqrt{2}=7(2+\sqrt{2})$ 

So, option D is correct.

Multiple choice

What is the name of the theorem that states that the area of a triangle is equal to half the product of its base and height?

  1. Brahmagupta's theorem

  2. Pythagorean theorem

  3. Euler's theorem

  4. Heron's formula
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Heron's formula provides a method for calculating the area of a triangle using its side lengths.

Multiple choice

What is the name of the theorem that states that the area of a triangle is equal to half the product of its base and height?

  1. Brahmagupta's theorem

  2. Pythagorean theorem

  3. Euler's theorem

  4. Heron's formula
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Heron's formula provides a method for calculating the area of a triangle using its side lengths.

Multiple choice

What is the name of the theorem that states that the area of a triangle is equal to half the product of its base and height?

  1. Brahmagupta's Theorem

  2. Pythagorean Theorem

  3. Heron's Formula

  4. Euler's Formula
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Heron's Formula states that the area of a triangle is equal to half the product of its base and height.

Multiple choice

What is the name of the theorem that states that the area of a triangle is equal to half the product of its base and height?

  1. Brahmagupta's Theorem

  2. Pythagorean Theorem

  3. Heron's Formula

  4. Euler's Formula
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Heron's Formula states that the area of a triangle is equal to half the product of its base and height.

Multiple choice

What is the name of the theorem that states that the area of a triangle is equal to half the product of its base and height?

  1. Pythagorean theorem

  2. Euclid's theorem

  3. Brahmagupta's theorem

  4. Bhaskara's theorem
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Brahmagupta's theorem states that the area of a triangle is equal to half the product of its base and height.

Multiple choice

The ancient Indian mathematical treatise, the 'Aryabhatiya', contains a formula for calculating the area of a triangle. What is the formula?

  1. Area = (1/2) * base * height

  2. Area = base * height

  3. Area = (1/2) * base^2

  4. Area = base^2 * height
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Aryabhatiya provides a formula for calculating the area of a triangle, which is Area = (1/2) * base * height.

Multiple choice

What is the area of a triangle with sides of length 3, 4, and 5?

  1. 6

  2. 8

  3. 10

  4. 12
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

To find the area of a triangle with sides of length 3, 4, and 5, we can use Heron's formula: (Area = \sqrt{s(s-a)(s-b)(s-c)}), where s is the semiperimeter of the triangle. The semiperimeter is (s = \frac{a + b + c}{2} = \frac{3 + 4 + 5}{2} = 6). Substituting the values of s, a, b, and c into the formula, we get: (Area = \sqrt{6(6-3)(6-4)(6-5)} = \sqrt{6 \cdot 3 \cdot 2 \cdot 1} = \sqrt{36} = 6). Therefore, the area of the triangle is 6.

Multiple choice

What is the name of the theorem that states that the area of a triangle is equal to half the product of its base and height?

  1. Pythagorean Theorem

  2. Triangle Inequality Theorem

  3. Angle Sum Theorem

  4. Heron's Formula
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Heron's Formula states that the area of a triangle is equal to half the product of its base and height. This formula is attributed to Indian mathematicians.

Multiple choice

The area of a triangle is given by the formula:

  1. (1/2) * base * height

  2. (1/2) * base * hypotenuse

  3. base * height
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The area of a triangle is given by the formula (1/2) * base * height.

Multiple choice

What is the name of the Chinese theorem that states that the area of a triangle is equal to half the product of its base and height?

  1. The Pythagorean Theorem

  2. The Triangle Inequality

  3. The Law of Cosines

  4. Heron's Formula
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Heron's Formula states that the area of a triangle is equal to half the product of its base and height. This formula was first discovered by Chinese mathematicians in the 1st century CE.

Multiple choice

In a triangle, the ratio of the lengths of two sides is 3:4. If the perimeter of the triangle is 42 cm, what is the length of the third side?

  1. 10 cm

  2. 12 cm

  3. 14 cm

  4. 16 cm
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let the lengths of the two sides be 3x and 4x. Then, the third side has length 42 - 3x - 4x = 42 - 7x. Since the perimeter of the triangle is 42 cm, we have 3x + 4x + 42 - 7x = 42. Solving for x, we get x = 6. Therefore, the length of the third side is 42 - 7x = 42 - 7 * 6 = 14 cm.