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High School Math : Triangles

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Example Questions

Example Question #31 : Triangles

Isosceles

The measure of the sides of this isosceles right triangle are $x=3\sqrt{2}$ . Find the measure of its hypotenuse, .

Possible Answers:

Not enough information to solve

Correct answer:

Explanation:

An isosceles triangle is a special triangle due to the values of its angles. These triangles are referred to as $45^{\circ}-45^{\circ}-90^{\circ}$ triangles and their side lenghts follow a specific pattern that states you can calculate the length of the hypotenuse of an isoceles triangle by multiplying the length of one of the legs by the square root of 2.

$h=x\sqrt{2}$

$h=3\sqrt{2}*\sqrt{2}$

h=3*2

$\rightarrow6$

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Example Question #1 : Acute / Obtuse Triangles

Two similiar triangles have a ratio of perimeters of $7:2$ .

If the smaller triangle has sides of 3, 7, and 5, what is the perimeter of the larger triangle.

Possible Answers:

$49.5$

$50.5$

$51.5$

$52.5$

$48.5$

Correct answer:

$52.5$

Explanation:

Adding the sides gives a perimeter of 15 for the smaller triangle. Multipying by the given ratio of $\frac{7}{2}$ , yields 52.5.

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Example Question #2 : Isosceles Triangles

Two sides of an isosceles triangle are 20 and 30. What is the difference of the largest and the smallest possible perimeters?

Possible Answers:

The answer cannot be determined

Correct answer:

Explanation:

The trick here is that we don't know which is the repeated side. Our possible triangles are therefore 20 + 20 + 30 = 70 or 30 + 30 + 20 = 80. The difference is therefore 80 – 70 or 10.

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Example Question #1 : Acute / Obtuse Triangles

Two similiar triangles exist where the ratio of perimeters is 4:5 for the smaller to the larger triangle. If the larger triangle has sides of 6, 7, and 12 inches, what is the perimeter, in inches, of the smaller triangle?

Possible Answers:

Correct answer:

Explanation:

The larger triangle has a perimeter of 25 inches. Therefore, using a 4:5 ratio, the smaller triangle's perimeter will be 20 inches.

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Example Question #1 : Acute / Obtuse Triangles

If a = 7 and b = 4, which of the following could be the perimeter of the triangle?

Msp30320bea9chb26d18hh0000627i1e4ibba8c4f5

I. 11

II. 15

III. 25

Possible Answers:

I Only

II and III Only

I, II and III

II Only

I and II Only

Correct answer:

II Only

Explanation:

Consider the perimeter of a triangle:

P = a + b + c

Since we know a and b, we can find c.

In I:

11 = 7 + 4 + c

11 = 11 + c

c = 0

Note that if c = 0, the shape is no longer a trial. Thus, we can eliminate I.

In II:

15 = 7 + 4 + c

15 = 11 + c

c = 4.

This is plausible given that the other sides are 7 and 4.

In III:

25 = 7 + 4 + c

25 = 11 + c

c = 14.

It is not possible for one side of a triangle to be greater than the sum of both of the other sides, so eliminate III.

Thus we are left with only II.

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

Solve for . (Not drawn to scale).

Possible Answers:

57^o

22^o

81^o

53^o

Correct answer:

53^o

Explanation:

The angles of a triangle must add to 180^o. In the triangle to the right, we know one angle and can find another using supplementary angles.

57+(180-110)+x=180

Now we only need to solve for .

57+70+x=180

x=180-(57+70)=180-127

x=53^o

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

Exterior_angle If $\angle A=42^{\circ}$ and $\angle B=76^{\circ}$ , what is the measure of $\theta$ ?

Possible Answers:

$62^{\circ}$

$76^{\circ}$

$118^{\circ}$

$42^{\circ}$

Not enough information to solve

Correct answer:

$118^{\circ}$

Explanation:

All of the interior angles of a triangle add up to $180^{\circ}$ .

If $\angle A=42^{\circ}$ and $\angle B=76^{\circ}$ , then

$\angle C= 180^{\circ}-(42^{\circ}+76^{\circ})$

Therefore, $\angle C=62^{\circ}$

Now, $\theta$ will equal $180^{\circ}-\angle C$ because $\angle C$ and $\theta$ form a straight line. Therefore,

$\theta=180^{\circ}-62^{\circ}$

$\rightarrow118^{\circ}$

Also, by definition, the angle of an exterior angle of a triangle is equal to the measure of the two interior angles opposite of it $\dpi{100} (\theta =76^{\circ}+42^{\circ}\rightarrow 118^{\circ})$ .

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Example Question #2 : Acute / Obtuse Triangles

Two interior angles in an obtuse triangle measure $123^{\circ}$ and $11^{\circ}$ . What is the measurement of the third angle.

Possible Answers:

$123^{\circ}$

$46^{\circ}$

$104^{\circ}$

$50^{\circ}$

$57^{\circ}$

Correct answer:

$46^{\circ}$

Explanation:

Interior angles of a triangle always add up to 180 degrees.

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Example Question #1 : Acute / Obtuse Triangles

In a given triangle, the angles are in a ratio of 1:3:5. What size is the middle angle?

Possible Answers:

$45^{\circ}$

$60^{\circ}$

$75^{\circ}$

$20^{\circ}$

$90^{\circ}$

Correct answer:

$60^{\circ}$

Explanation:

Since the sum of the angles of a triangle is $180^{\circ}$ , and given that the angles are in a ratio of 1:3:5, let the measure of the smallest angle be , then the following expression could be written:

$x+3x+5x=180$

$9x=180$

$x=20$

If the smallest angle is 20 degrees, then given that the middle angle is in ratio of 1:3, the middle angle would be 3 times as large, or 60 degrees.

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Example Question #4 : Isosceles Triangles

Triangle ABC has angle measures as follows:

$\dpi{100} \small m\angle ABC=4x+3$

$\dpi{100} \small m\angle ACB=2x+6$

$\dpi{100} \small m\angle BAC=3x$

What is $\dpi{100} \small m\angle BAC$ ?

Possible Answers:

Correct answer:

Explanation:

The sum of the measures of the angles of a triangle is 180.

Thus we set up the equation $\dpi{100} \small 4x+3+2x+6+3x=180$

After combining like terms and cancelling, we have $\dpi{100} \small 9x=171\rightarrow x=19$

Thus $\dpi{100} \small m\angle BAC=3x=57$

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High School Math : Triangles

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #31 : Triangles

Example Question #1 : Acute / Obtuse Triangles

Example Question #2 : Isosceles Triangles

Example Question #1 : Acute / Obtuse Triangles

Example Question #1 : Acute / Obtuse Triangles

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

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

Example Question #2 : Acute / Obtuse Triangles

Example Question #1 : Acute / Obtuse Triangles

Example Question #4 : Isosceles Triangles

All High School Math Resources

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