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

Two triangles have corresponding sides of 7 cm, 24 cm, and 25 cm, and 14 cm, 48 cm, and 50 cm respectively. Are the triangles similar?

  1. Yes

  2. No
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

To determine if the triangles are similar, we need to check if the ratios of their corresponding sides are equal. The ratios are 7/14 = 1/2, 24/48 = 1/2, and 25/50 = 1/2. Since all the ratios are equal, the triangles are similar.

Multiple choice

If two similar triangles have a scale factor of 4:5, what is the ratio of the area of the smaller triangle to the area of the larger triangle?

  1. 4:5

  2. 5:4

  3. 16:25

  4. 25:16
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The ratio of the areas of similar triangles is equal to the square of the scale factor. Therefore, the ratio of the area of the smaller triangle to the area of the larger triangle is (4/5)^2 = 16/25.

Multiple choice

In a triangle ABC, the side AB is 10 cm and the side AC is 12 cm. If the triangle is similar to triangle DEF, where DE is 15 cm, what is the length of DF?

  1. 18 cm

  2. 20 cm

  3. 22 cm

  4. 24 cm
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Since the triangles are similar, the ratio of their corresponding sides is equal. Therefore, AB/DE = AC/DF. Substituting the given values, we get 10/15 = 12/DF. Solving for DF, we get DF = 20 cm.

Multiple choice

Two triangles have corresponding sides of 9 cm, 15 cm, and 20 cm, and 18 cm, 30 cm, and 40 cm respectively. Are the triangles similar?

  1. Yes

  2. No
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

To determine if the triangles are similar, we need to check if the ratios of their corresponding sides are equal. The ratios are 9/18 = 1/2, 15/30 = 1/2, and 20/40 = 1/2. Since all the ratios are equal, the triangles are similar.

Multiple choice

What is the formula for the surface area of a pyramid with base perimeter (p) and slant height (l)?

  1. \(\frac{1}{2}pl\)
  2. \(pl\)
  3. \(2pl\)
  4. \(3pl\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The surface area of a pyramid is given by the formula (\frac{1}{2}pl), where (p) is the perimeter of the base and (l) is the slant height of the pyramid.

Multiple choice

What is the formula for the surface area of a triangular prism with base perimeter (p) and slant height (l)?

  1. \(pl\)
  2. \(2pl\)
  3. \(3pl\)
  4. \(4pl\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The surface area of a triangular prism is given by the formula (pl), where (p) is the perimeter of the base and (l) is the slant height of the prism.

Multiple choice

What is the formula for the area of a triangle?

  1. $A = \frac{1}{2}bh$
  2. $A = bh$
  3. $A = \frac{1}{2}b^2h$
  4. $A = b^2h$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The formula for the area of a triangle is $A = \frac{1}{2}bh$, where $b$ is the length of the base and $h$ is the height (altitude) of the triangle.

Multiple choice

What is the formula for the perimeter of a triangle?

  1. $P = a + b + c$
  2. $P = 2a + 2b + 2c$
  3. $P = a^2 + b^2 + c^2$
  4. $P = 2(a + b + c)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The formula for the perimeter of a triangle is $P = a + b + c$, where $a$, $b$, and $c$ are the lengths of the three sides of the triangle.

Multiple choice

What is the area of a triangle with sides of length 5, 12, and 13?

  1. 30 square units

  2. 60 square units

  3. 90 square units

  4. 120 square units
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Using Heron's formula, the area of the triangle is $\sqrt{s(s-a)(s-b)(s-c)}$, where $s = \frac{a+b+c}{2}$. Plugging in the values, we get $s = \frac{5+12+13}{2} = 15$, and the area is $\sqrt{15(15-5)(15-12)(15-13)} = 60$ square units.

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. Pythagoras' Theorem

  2. Euler's Formula

  3. Bhaskara's Formula

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

Heron's Formula is a mathematical theorem that states that the area of a triangle is equal to half the product of its base and height. It was discovered by the Greek mathematician Heron of Alexandria in the 1st century AD. Heron's Formula is widely used in geometry and has applications in various fields, including surveying, engineering, and architecture.

Multiple choice

What is the formula for the area of a triangle, as given by Bhaskara II?

  1. $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$
  2. $\text{Area} = \frac{1}{2} \times \text{base} \times \text{side}$
  3. $\text{Area} = \frac{1}{2} \times \text{side} \times \text{height}$
  4. $\text{Area} = \frac{1}{2} \times \text{side} \times \text{side}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Bhaskara II's formula for the area of a triangle is $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$. This formula is still used today in geometry.

Multiple choice

What was the Babylonian method for calculating the area of a triangle?

  1. The Babylonian Method

  2. The Heron's Formula

  3. The Pythagorean Theorem

  4. The Area of a Triangle Formula
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Babylonians had a method for calculating the area of a triangle that involved using a series of approximations to find the solution.

Multiple choice

Find the area of a triangle with sides of length 6 cm, 8 cm, and 10 cm.

  1. 24 cm^2

  2. 36 cm^2

  3. 48 cm^2

  4. 60 cm^2
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

To find the area of a triangle, we can use Heron's formula: (Area = \sqrt{s(s-a)(s-b)(s-c)}), where (s) is the semi-perimeter and (a, b, c) are the lengths of the sides. Plugging in the values of (a = 6 cm, b = 8 cm, c = 10 cm), we get: (s = \frac{6 cm + 8 cm + 10 cm}{2} = 12 cm). Therefore, the area of the triangle is: (Area = \sqrt{12 cm(12 cm - 6 cm)(12 cm - 8 cm)(12 cm - 10 cm)} = 36 cm^2).

Multiple choice

Find the area of the triangle formed by the lines (y = 2x + 1), (y = x - 1), and (x = 0).

  1. \(2\) square units
  2. \(3\) square units
  3. \(4\) square units
  4. \(5\) square units
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

To find the area of the triangle, we can use the formula: (Area = \frac{1}{2}|x_1 y_2 - x_2 y_1|), where ((x_1, y_1)) and ((x_2, y_2)) are the coordinates of two vertices of the triangle. Plugging in the coordinates of the vertices, we get: (Area = \frac{1}{2}|(0)(2) - (0)(-1)| = \frac{1}{2}|0 - 0| = \frac{1}{2}(0) = 0) square units.

Multiple choice

What is the area of the triangle formed by the points (1, 2), (3, 4), and (5, 2)?

  1. 6 square units

  2. 8 square units

  3. 10 square units

  4. 12 square units
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

We can use Heron's formula to find the area of the triangle. Heron's formula states that the area of a triangle with sides a, b, and c is given by the formula √(s(s-a)(s-b)(s-c)), where s is the semiperimeter of the triangle. In this case, the semiperimeter of the triangle is (1 + 3 + 5)/2 = 4.5. Plugging this value into Heron's formula, we get the area of the triangle as √(4.5(4.5-1)(4.5-3)(4.5-5)) = √(4.5(3.5)(1.5)(-0.5)) = √(10.125) = 10 square units.