Mathematics ยท 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

Let ABC be a triangle with sides AB = 13, BC = 14, and CA = 15. Let D be a point on BC such that BD = 5. Let E be a point on CA such that CE = 4. Let F be a point on AB such that AF = 3. Find the area of triangle DEF.

  1. 12

  2. 18

  3. 24

  4. 30
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The area of triangle DEF is equal to the area of triangle ABC minus the areas of triangles ABD, BCE, and ACF. The area of triangle ABC is (1/2)(13)(14) = 91. The area of triangle ABD is (1/2)(5)(12) = 30. The area of triangle BCE is (1/2)(4)(9) = 18. The area of triangle ACF is (1/2)(3)(12) = 18. Therefore, the area of triangle DEF is 91 - 30 - 18 - 18 = 24.

Multiple choice

Let $ABC$ be a triangle with sides $AB = 13$, $BC = 14$, and $CA = 15$. Let $D$ be a point on $BC$ such that $BD = 5$. Let $E$ be a point on $CA$ such that $CE = 4$. Let $F$ be a point on $AB$ such that $AF = 3$. Find the area of triangle $DEF$.

  1. 12

  2. 18

  3. 24

  4. 30
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The area of triangle $DEF$ is equal to the area of triangle $ABC$ minus the areas of triangles $ABD$, $BCE$, and $ACF$. The area of triangle $ABC$ is $rac{1}{2}(13)(14) = 91$. The area of triangle $ABD$ is $rac{1}{2}(5)(12) = 30$. The area of triangle $BCE$ is $rac{1}{2}(4)(9) = 18$. The area of triangle $ACF$ is $rac{1}{2}(3)(12) = 18$. Therefore, the area of triangle $DEF$ is $91 - 30 - 18 - 18 = 24$.

Multiple choice

Let $ABC$ be a triangle with sides $AB = 13$, $BC = 14$, and $CA = 15$. Let $D$ be a point on $BC$ such that $BD = 5$. Let $E$ be a point on $CA$ such that $CE = 4$. Let $F$ be a point on $AB$ such that $AF = 3$. Find the area of triangle $DEF$.

  1. 12

  2. 18

  3. 24

  4. 30
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The area of triangle $DEF$ is equal to the area of triangle $ABC$ minus the areas of triangles $ABD$, $BCE$, and $ACF$. The area of triangle $ABC$ is $rac{1}{2}(13)(14) = 91$. The area of triangle $ABD$ is $rac{1}{2}(5)(12) = 30$. The area of triangle $BCE$ is $rac{1}{2}(4)(9) = 18$. The area of triangle $ACF$ is $rac{1}{2}(3)(12) = 18$. Therefore, the area of triangle $DEF$ is $91 - 30 - 18 - 18 = 24$.

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. Heron's formula

  2. Pythagorean theorem

  3. Brahmagupta's theorem

  4. Triangle inequality theorem
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Heron's formula is a well-known result in geometry that 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. Heron's formula

  2. Pythagorean theorem

  3. Brahmagupta's theorem

  4. Triangle area formula
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The triangle area formula is a well-known result in geometry that provides a method for calculating the area of a triangle using its base and height.

Multiple choice

Find the area of a triangle with a base of 10 cm and a height of 8 cm.

  1. 40 sq cm

  2. 80 sq cm

  3. 120 sq cm

  4. 160 sq cm
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The area of a triangle is given by the formula A = (1/2) * base * height. Substituting the given values, we get A = (1/2) * 10 cm * 8 cm = 40 sq cm.

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. Euler's theorem

  3. Fermat's Last 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.

Multiple choice

Which theorem 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. Area of a Triangle Theorem

  4. Law of Cosines
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The Area of a Triangle Theorem 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 mathematical theorem that states that the area of a triangle is equal to half the product of its base and height?

  1. Pythagorean Theorem

  2. Pascal's Triangle

  3. Euler's Formula

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

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

Multiple choice

In a triangle ABC, the sides AB and AC are 6 cm and 8 cm respectively. If the triangle is similar to triangle DEF, where DE is 9 cm, what is the length of DF?

  1. 12 cm

  2. 15 cm

  3. 18 cm

  4. 21 cm
Reveal answer Fill a bubble to check yourself
C 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 6/9 = 8/DF. Solving for DF, we get DF = 18 cm.

Multiple choice

If two similar triangles have a scale factor of 3:5, what is the ratio of the area of the smaller triangle to the area of the larger triangle?

  1. 3:5

  2. 5:3

  3. 9:25

  4. 25:9
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 (3/5)^2 = 9/25.

Multiple choice

In a triangle ABC, the side AB is 12 cm and the side AC is 15 cm. If the triangle is similar to triangle DEF, where DE is 18 cm, what is the length of DF?

  1. 22.5 cm

  2. 27 cm

  3. 30 cm

  4. 36 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 12/18 = 15/DF. Solving for DF, we get DF = 27 cm.

Multiple choice

Two triangles have corresponding sides of 5 cm, 12 cm, and 13 cm, and 10 cm, 24 cm, and 26 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 5/10 = 1/2, 12/24 = 1/2, and 13/26 = 1/2. Since all the ratios are equal, the triangles are similar.

Multiple choice

If two similar triangles have a scale factor of 2:3, what is the ratio of the perimeter of the smaller triangle to the perimeter of the larger triangle?

  1. 2:3

  2. 3:2

  3. 4:9

  4. 9:4
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The ratio of the perimeters of similar triangles is equal to the scale factor. Therefore, the ratio of the perimeter of the smaller triangle to the perimeter of the larger triangle is 2:3.

Multiple choice

In a triangle ABC, the side AB is 8 cm and the side AC is 10 cm. If the triangle is similar to triangle DEF, where DE is 12 cm, what is the length of DF?

  1. 15 cm

  2. 18 cm

  3. 21 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 8/12 = 10/DF. Solving for DF, we get DF = 18 cm.