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Intermediate Geometry : Triangles

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Example Question #81 : Triangles

A triangle is defined by the following points in a coordinate plane: (-1, -1), (-2, -2), (-5, 0) .

What is the perimeter of the triangle?

Possible Answers:

11.44

8.55

12.09

9.14

Correct answer:

9.14

Explanation:

In order to find the perimeter of the triangle, we will first need to find the length of each side of the triangle by using the distance formula.

Recall the distance formula for a line:

$\text{distance}=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

The first side of the triangle is the line segment made with $(-1, -1)\text{ and }(-2, -2)$ as its endpoints.

$\text{side 1}=\sqrt{(-2-(-1))^2+(-2-(-1))^2}=\sqrt{2}$

The second side of the triangle is the line segment that has $(-2, -2)\text{ and }(-5, 0)$ as its endpoints.

$\text{side 2}=\sqrt{(-5-(-2))^2+(0-(-2))^2}=\sqrt{13}$

The third side of the triangle is the line segment that has $(-1, -1)\text{ and }(-5, 0)$ as its endpoints.

$\text{side 3}=\sqrt{(-5-(-1))^2+(0-(-1))^2}=\sqrt{17}$

Now, add up these three sides with a calculator to find the perimeter of the triangle.

$\text{Perimeter}=\sqrt{2} +\sqrt{13}+\sqrt{17}=9.14$

Make sure to round to places after the decimal.

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Example Question #1 : How To Find If Acute / Obtuse Triangles Are Congruent

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

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

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

$\angle C \cong \angle F$

True or false: It follows from the given information that $\bigtriangleup ABC \cong \bigtriangleup DEF$ .

Possible Answers:

True

False

Correct answer:

False

Explanation:

Examine the diagram below.

Untitled

$\overline{AB} \cong \overline{DE}$ , $\overline{AC} \cong \overline{DF}$ , and $\angle C \cong \angle F$ , but $\overline{BC}\ncong \overline{EF}$ . As a result, it is not true that $\bigtriangleup ABC \cong \bigtriangleup DEF$ . Therefore, the statement is false.

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Example Question #1 : How To Find If Acute / Obtuse Triangles Are Congruent

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

$\angle A \cong \angle D$

$\angle B \cong \angle E$

$\angle C \cong \angle F$

True or false: It follows from the information given that $\bigtriangleup ABC \cong \bigtriangleup DEF$ .

Possible Answers:

False

True

Correct answer:

False

Explanation:

The congruence of corresponding angles of two triangles does not alone prove that the triangles are congruent. For example, see the figures below:

Triangles 1

The three angle congruence statements are true, but the sides are not congruent, so the triangles are not congruent. The statement is false.

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Example Question #3 : How To Find If Acute / Obtuse Triangles Are Congruent

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

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

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

$\overline{BC} \cong \overline{EF}$

True or false: It follows from the given information that $\bigtriangleup ABC \cong \bigtriangleup DEF$ .

Possible Answers:

False

True

Correct answer:

True

Explanation:

As we are establishing whether or not $\bigtriangleup ABC \sim \bigtriangleup DEF$ , then , , and correspond respectively to , , and .

By the Side-Side-Side Congruence Postulate (SSS), if all three pairs of corresponding sides of two triangles are congruent, then the triangles themselves are congruent. Between $\bigtriangleup ABC$ and $\bigtriangleup DEF$ , $\overline{AB}$ and $\overline{DE}$ are corresponding sides, their congruence is given. The other two congruences between corresponding sides are given, so the conditions of SSS are satisfied. $\bigtriangleup ABC \cong \bigtriangleup DEF$ is indeed true.

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Example Question #4 : How To Find If Acute / Obtuse Triangles Are Congruent

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

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

$\overline{AC} \cong \overline{EF}$

$\overline{BC}\cong \overline{DF}$

True or false: It follows from the given information that $\bigtriangleup ABC \cong \bigtriangleup DEF$ .

Possible Answers:

True

False

Correct answer:

False

Explanation:

As we are establishing whether or not $\bigtriangleup ABC \sim \bigtriangleup DEF$ , then , , and correspond respectively to , , and .

By the Side-Side-Side Congruence Postulate (SSS), if all three pairs of corresponding sides of two triangles are congruent, then the triangles themselves are congruent. However, if we restate the first side congruence as

$\overline{AB} \cong \overline{ED}$

and examine it with the other two:

$\overline{AC} \cong \overline{EF}$

$\overline{BC}\cong \overline{DF}$

We see that while we can invoke SSS, the points A,B, C correspond to E,D,F , respectively. The triangle congruence that follows is therefore $\bigtriangleup ABC \cong \bigtriangleup EDF$ .

The answer is therefore false.

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Example Question #4 : How To Find If Acute / Obtuse Triangles Are Congruent

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

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

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

$\angle A \cong \angle D$

True or false: It follows from the given information that $\bigtriangleup ABC \cong \bigtriangleup DEF$ .

Possible Answers:

False

True

Correct answer:

True

Explanation:

As we are establishing whether or not $\bigtriangleup ABC \cong \bigtriangleup DEF$ , then , , and correspond respectively to , , and .

By the Side-Angle-Side Congruence Postulate (SAS), if two pairs of corresponding sides and the included angle of one triangle are congruent to the corresponding parts of a second, the triangles are congruent. $\overline{AB} \cong \overline{DE}$ and $\overline{AC} \cong \overline{DF}$ , indicating congruence between corresponding sides, and $\angle A \cong \angle D$ , indicating congruence between corresponding included angles. This satisfies the conditions of SAS, so $\bigtriangleup ABC \cong \bigtriangleup DEF$ is true.

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Example Question #6 : How To Find If Acute / Obtuse Triangles Are Congruent

Hinge

Refer to the above two triangles. By what statement does it follow that $\bigtriangleup ABC \cong \bigtriangleup DEF$ ?

Possible Answers:

The Hinge Theorem

The Angle-Angle-Side Theorem

The Angle-Side-Angle Postulate

The Converse of the Isosceles Triangle Theorem

The Converse of the Pythagorean Theorem

Correct answer:

The Angle-Angle-Side Theorem

Explanation:

We are given that two angles of $\bigtriangleup ABC$ - $\angle B$ and $\angle C$ - and a nonincluded side $\overline{AC}$ are congruent to their corresponding parts, $\angle E$ , $\angle F$ , and $\overline{DF}$ of $\bigtriangleup DEF$ . It follows from the Angle-Angle-Side Theorem that $\bigtriangleup ABC \cong \bigtriangleup DEF$ .

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Example Question #7 : How To Find If Acute / Obtuse Triangles Are Congruent

Hinge

Refer to the above two triangles. By what statement does it follow that $\bigtriangleup ABC \cong \bigtriangleup DEF$ ?

Possible Answers:

The Hinge Theorem

The Triangle Midsegment Theorem

The Side-Angle-Side Postulate

The Isosceles Triangle Theorem

The Angle-Angle Postulate

Correct answer:

The Side-Angle-Side Postulate

Explanation:

We are given that two sides of $\bigtriangleup ABC$ - sides $\overline{AB}$ and $\overline{BC}$ - and their included angle $\angle B$ are congruent to their corresponding parts, sides $\overline{DE}$ and $\overline{EF}$ and $\angle E$ of $\bigtriangleup DEF$ . It follows from the Side-Angle-Side Postulate that $\bigtriangleup ABC \cong \bigtriangleup DEF$ .

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Example Question #1 : How To Find An Angle In An Acute / Obtuse Triangle

In ΔABC, A = 75°, a = 13, and b = 6.

Find B (to the nearest tenth).

Possible Answers:

30.4°

28.1°

34.9°

26.5°

27.8°

Correct answer:

26.5°

Explanation:

This problem requires us to use either the Law of Sines or the Law of Cosines. To figure out which one we should use, let's write down all the information we have in this format:

A = 75° a = 13

B = ? b = 6

C = ? c = ?

Now we can easily see that we have a complete pair, A and a. This tells us that we can use the Law of Sines. (We use the Law of Cosines when we do not have a complete pair).

Law of Sines:
$\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$

To solve for b, we can use the first two terms which gives us:
$\frac{\sin 75°}{13}=\frac{\sin B°}{6}$
$\sin B =6*\frac{\sin 75}{13}$
B=26.5°

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Example Question #2 : Acute / Obtuse Triangles

Points A, B, and C are collinear (they lie along the same line). The measure of angle CAD is $30^{\circ}$ . The measure of angle CBD is $60^{\circ}$ . The length of segment $\overline{AD}$ is 4.

Find the measure of $\dpi{100} \small \angle ADB$ .

Possible Answers:

$30^{\circ}$

$90^{\circ}$

$60^{\circ}$

$15^{\circ}$

$45^{\circ}$

Correct answer:

$30^{\circ}$

Explanation:

The measure of $\dpi{100} \small \angle ADB$ is $30^{\circ}$ . Since $\dpi{100} \small A$ , $\dpi{100} \small B$ , and $\dpi{100} \small C$ are collinear, and the measure of $\dpi{100} \small \angle CBD$ is $60^{\circ}$ , we know that the measure of $\dpi{100} \small \angle ABD$ is $120^{\circ}$ .

Because the measures of the three angles in a triangle must add up to $180^{\circ}$ , and two of the angles in triangle $\dpi{100} \small ABD$ are $30^{\circ}$ and $120^{\circ}$ , the third angle, $\dpi{100} \small \angle ADB$ , is $30^{\circ}$ .

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Intermediate Geometry : Triangles

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #81 : Triangles

Example Question #1 : How To Find If Acute / Obtuse Triangles Are Congruent

Example Question #1 : How To Find If Acute / Obtuse Triangles Are Congruent

Example Question #3 : How To Find If Acute / Obtuse Triangles Are Congruent

Example Question #4 : How To Find If Acute / Obtuse Triangles Are Congruent

Example Question #4 : How To Find If Acute / Obtuse Triangles Are Congruent

Example Question #6 : How To Find If Acute / Obtuse Triangles Are Congruent

Example Question #7 : How To Find If Acute / Obtuse Triangles Are Congruent

Example Question #1 : How To Find An Angle In An Acute / Obtuse Triangle

Example Question #2 : Acute / Obtuse Triangles

All Intermediate Geometry Resources

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