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Basic Geometry : Basic Geometry

Study concepts, example questions & explanations for Basic Geometry

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9 Diagnostic Tests 164 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1481 : Plane Geometry

Are these right triangles congruent?

Congruent right triangles

Possible Answers:

Cannot be determined - we need at least one pair of angles, or all three sides

No - at least one pair of corresponding sides is not congruent

No - the angles are different

Yes - all three pairs of sides must be congruent by Pythagorean Theorem

Yes - by the angle-angle-side theorem

Correct answer:

Yes - all three pairs of sides must be congruent by Pythagorean Theorem

Explanation:

Right now we can't directly compare these triangles because we do not know all three side lengths. However, we can use Pythagorean Theorem to determine both missing sides. The left triangle is missing the hypotenuse:

$\small 48^2 + 55^2 = c^2$

$\small 2,304 + 3,025 = c^2$

$\small 5,329 = c$

$\small c = \sqrt{5,329}= 73$

The right triangle is missing one of the legs:

$\small x^2 + 48^2 = 78^2$

$\small x^2 + 2,304 = 5,329$ subtract 2,304 from both sides

$\small x^2 = 3,025$

$\small x = \sqrt{3,025} = 55$

This means that the two triangles both have side lengths 48, 55, 73, so they must be congruent.

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Example Question #1481 : Basic Geometry

The hypotenuse and acute angle are given for several triangles. Which if any are congruent? Triangle A- Hypotenuse=15; acute angle=56 degrees. Triangle B- Hypotenuse=18; acute angle=56 degrees. Triangle C-Hypotenuse=18; acute angle= 45 degrees.

Possible Answers:

A & C

All three.

B & C

None of these

A & B

Correct answer:

None of these

Explanation:

The correct answer is none of these. There are several pairs of angles and sides or sides and angles that must be the same in order for two triangles to be congruent.

In our case, we need the acute angle and the hypotenuse to both be equal. No two triangles above have this relationship and therefore no two are congruent.

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Example Question #1483 : Basic Geometry

Given: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ .

$\angle B$ and $\angle E$ are both right angles.

$\overline{AB } \cong \overline{DE}$

$\overline{AC} \cong \overline{DF}$

True or false: From the above information, it follows that $\bigtriangleup ABC \cong \bigtriangleup DEF$ .

Possible Answers:

True

False

Correct answer:

True

Explanation:

If we seek to prove that $\bigtriangleup ABC \cong \bigtriangleup DEF$ , then , , and correspond to , , and , respectively.

By the Hypotenuse-Leg Theorem (HL), if the hypotenuse and one leg of a triangle are congruent to those of another, the triangles are congruent.

$\angle B$ and $\angle E$ are both right angles, so $\bigtriangleup ABC$ and $\bigtriangleup DEF$ are both right triangles. $\overline{AB }$ and $\overline{DE}$ are congruent corresponding sides, and moreover, since, each includes the right-angle vertex as an endpoint, they are congruent corresponding legs. $\overline{AC}$ and $\overline{DF}$ are opposite the right angles, making them congruent corresponding hypotenuses.

The conditions of HL are satisfied, so $\bigtriangleup ABC \cong \bigtriangleup DEF$ .

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Example Question #302 : Right Triangles

Given: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ .

$\angle B$ and $\angle E$ are both right angles.

$\overline{AB} \cong \overline{BC}$

$\overline{DE} \cong \overline{EF}$

True or false: From the given information, it follows that $\bigtriangleup ABC \cong \bigtriangleup DEF$ .

Possible Answers:

False

True

Correct answer:

False

Explanation:

The congruence of $\bigtriangleup ABC$ and $\bigtriangleup DEF$ cannot be proved from the given information alone. Examine the two triangles below:

Triangles 2

$\overline{AB} \cong \overline{BC}$ , $\overline{DE} \cong \overline{EF}$ , and $\angle B$ and $\angle E$ are both right angles, so the conditions of the problem are met; however, since the sides are not congruent between triangles - for example, $\overline{AB} \ncong \overline{DE}$ - the triangles are not congruent either.

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Example Question #1 : How To Find The Height Of A Right Triangle

A right triangle has a hypotenuse of 18 inches and a base of 12 inches. What is the height of the triangle in inches?

Possible Answers:

180.2

13.42

8.73

12.55

Correct answer:

13.42

Explanation:

We can find one leg of a right triangle when we have the length of the hypotenuse and the other leg. Square the hypotenuse and the known leg. Then, subtract the squared length of the leg from the squared length of the hypotenuse. Finally, find the square root of the result.

$\sqrt{18^{2}-12^{2}}=13.42$

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Example Question #1 : How To Find The Height Of A Right Triangle

Screen_shot_2015-02-27_at_6.56.15_pm

If c is not the base of the triangle, which of the following is the height?

Possible Answers:

None of the other answers

Correct answer:

Explanation:

If is not the base, that makes either or the base. If either or is the base, the right angle is on the bottom, so or respectively will be perpendicular. The height of a triangle is the distance from the base to the highest point, and in a right triangle that will be found by the side adjoining the base at a right angle. So if the base is , then and vise versa.

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Example Question #1 : How To Find The Height Of A Right Triangle

The area of a right triangle is 28. If one leg has a length of 7, what is the length of the other leg?

Possible Answers:

Correct answer:

Explanation:

We begin with the formula for the area of a triangle.

$A=\frac{1}{2}bh$

We further realize that the two legs of the triangle are the base and height; therefore, substituting what we know we get

$28=\frac{1}{2}(7)h$

$28=\frac{7}{2}h$

We can then solve by simply dividing.

8 = h

We have found our height and thus the second leg of our triangle.

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Example Question #1 : How To Find The Height Of A Right Triangle

If the area of a right triangle is , and the base of the right triangle is , what is the height of the right triangle?

Possible Answers:

Correct answer:

Explanation:

Recall how to find the area of a right triangle:

$\text{Area}=\frac{1}{2}(\text{base}\times\text{height})$

Now, we are going to manipulate the equation to solve for height.

$2(\text{Area})=\text{base}\times\text{height}$

$\text{Height}=\frac{2(\text{Area})}{\text{base}}$

Now, plug in the information given by the question about the values of the area and base of the triangle to find the height.

$\text{Height}=\frac{2(16)}{4}=\frac{32}{4}=8$

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Example Question #1 : How To Find The Height Of A Right Triangle

If the area of a right triangle is , and the base of the triangle is , what is the height of the triangle?

Possible Answers:

Correct answer:

Explanation:

Recall how to find the area of a right triangle:

$\text{Area}=\frac{1}{2}(\text{base}\times\text{height})$

Now, we are going to manipulate the equation to solve for height.

$2(\text{Area})=\text{base}\times\text{height}$

$\text{Height}=\frac{2(\text{Area})}{\text{base}}$

Now, plug in the information given by the question about the values of the area and base of the triangle to find the height.

$\text{Height}=\frac{2(10)}{5}=\frac{20}{5}=4$

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Example Question #1 : How To Find The Height Of A Right Triangle

If the area of a right triangle is , and the base of the triangle is , what is the height of the triangle?

Possible Answers:

1.5

4.5

Correct answer:

Explanation:

Recall how to find the area of a right triangle:

$\text{Area}=\frac{1}{2}(\text{base}\times\text{height})$

Now, we are going to manipulate the equation to solve for height.

$2(\text{Area})=\text{base}\times\text{height}$

$\text{Height}=\frac{2(\text{Area})}{\text{base}}$

Now, plug in the information given by the question about the values of the area and base of the triangle to find the height.

$\text{Height}=\frac{2(15)}{10}=\frac{30}{10}=3$

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Basic Geometry : Basic Geometry

Study concepts, example questions & explanations for Basic Geometry

All Basic Geometry Resources

Example Questions

Example Question #1481 : Plane Geometry

Example Question #1481 : Basic Geometry

Example Question #1483 : Basic Geometry

Example Question #302 : Right Triangles

Example Question #1 : How To Find The Height Of A Right Triangle

Example Question #1 : How To Find The Height Of A Right Triangle

Example Question #1 : How To Find The Height Of A Right Triangle

Example Question #1 : How To Find The Height Of A Right Triangle

Example Question #1 : How To Find The Height Of A Right Triangle

Example Question #1 : How To Find The Height Of A Right Triangle

All Basic Geometry Resources

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