ACT Math : Plane Geometry

Study concepts, example questions & explanations for ACT Math

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Example Questions

Example Question #1 : Acute / Obtuse Triangles

Points A, B, C, D are collinear. The measure of ∠ DCE is 130° and of ∠ AEC is 80°. Find the measure of ∠ EAD.

Screen_shot_2013-03-18_at_3.27.08_pm

Possible Answers:

60°

80°

70°

50°

Correct answer:

50°

Explanation:

To solve this question, you need to remember that the sum of the angles in a triangle is 180°. You also need to remember supplementary angles. If you know what ∠ DCE is, you also know what ∠ ECA is. Hence you know two angles of the triangle, 180°-80°-50°= 50°. 

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

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:

60^{\circ}

45^{\circ}

30^{\circ}

15^{\circ}

90^{\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}.

Example Question #171 : Act Math

Observe the following image and answer the question below:

Triangles

Are triangles  and  congruent?

Possible Answers:

No

Yes

Maybe

Not enough information to decide.

Correct answer:

Yes

Explanation:

Two triangles are only congruent if all of their sides are the same length, and all of the corresponding angles are of the same degree. Luckily, we only need three of these six numbers to completely determine the others, as long as we have at least one angle and one side, and any other combination of the other numbers.

In this case, we have two adjacent angles and one side, directly across from one of our angles in both triangles. This can be called the AAS case. We can see from our picture that all of our angles match, and the two sides match as well. They're all in the same position relative to each other on the triangle, so that is enough information to say that the two triangles are congruent.

Example Question #1 : How To Find The Perimeter Of An Acute / Obtuse Triangle

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:

50.5

52.5

48.5

49.5

51.5

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.

Example Question #2 : 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:

20

25

18

23

Correct answer:

20

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.

Example Question #2 : How To Find The Perimeter Of An Acute / Obtuse Triangle

Two similar triangles' perimeters are in a ratio of . If the lengths of the larger triangle's sides are , , and , what is the perimeter of the smaller triangle?

Possible Answers:

Correct answer:

Explanation:

1. Find the perimeter of the larger triangle:

2. Use the given ratio to find the perimeter of the smaller triangle:

Cross multiply and solve:

Example Question #4 : How To Find The Perimeter Of An Acute / Obtuse Triangle

There are two similar triangles. Their perimeters are in a ratio of . If the perimeter of the smaller triangle is , what is the perimeter of the larger triangle?

Possible Answers:

Correct answer:

Explanation:

Use proportions to solve for the perimeter of the larger triangle:

Cross multiply and solve:

Example Question #171 : Act Math

Two similar triangles have perimeteres in the ratio . The sides of the smaller triangle measure , , and  respectively. What is the perimeter, in meters, of the larger triangle?

Possible Answers:

Correct answer:

Explanation:

Since the perimeter of the smaller triangle is , and since the larger triangle has a perimeter in the  ratio, we can set up the following identity, where  the perimeter of the larger triangle:

 

 

In cross multiplying this identity, we get . We can now solve for . Here, , so the perimeter of the larger triangle is .

Example Question #11 : Acute / Obtuse Triangles

_tri11

What is the value of  in the triangle above? Round to the nearest hundredth.

Possible Answers:

Cannot be calculated

Correct answer:

Explanation:

Begin by filling in the missing angle for your triangle. Since a triangle has a total of  degrees, you know that the missing angle is:

Draw out the figure:

_tri12

Now, to solve this, you will need some trigonometry. Use the Law of Sines to calculate the value:

Solving for , you get:

Rounding, this is .

Example Question #171 : Act Math

The base of a triangle is  and the area is .  The height of the triangle is then decreased by . What is the final area of the triangle? 

Possible Answers:

Correct answer:

Explanation:

The formula for the area of a triangle is

  .

If the area is equal to 48 cm2 and the base is 8 cm, then the initial height is: 

If 12 is decreased by 75% then

, and . The final height is 3 cm.

Therefore the final area is

.

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