High School Math : Geometry

Study concepts, example questions & explanations for High School Math

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Example Questions

Example Question #331 : Geometry

What is the third angle in a triangle with angles of \(\displaystyle 120\) degrees and \(\displaystyle 70\) degrees? 

Possible Answers:

\(\displaystyle -10\) degrees 

No such triangle can exist.

\(\displaystyle 10\) degrees 

\(\displaystyle 70\) degrees 

\(\displaystyle 20\) degrees 

Correct answer:

No such triangle can exist.

Explanation:

We know that the sum of the angles of a triangle must add up to \(\displaystyle 180\) degrees. The two given angles sum to \(\displaystyle 190\) degrees. Thus, a triangle cannot be formed.

Example Question #332 : Geometry

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 : Triangles

If the average (arithmetic mean) of two noncongruent angles of an isosceles triangle is \(\displaystyle 55^o\), which of the following is the measure of one of the angles of the triangle?

Possible Answers:

\(\displaystyle 50^o\)

\(\displaystyle 90^o\)

\(\displaystyle 30^o\)

\(\displaystyle 40^o\)

\(\displaystyle 45^o\)

Correct answer:

\(\displaystyle 40^o\)

Explanation:

Since the triangle is isosceles, we know that 2 of the angles (that sum up to 180) must be equal. The question states that the noncongruent angles average 55°, thus providing us with a system of two equations:

\(\displaystyle \frac{x+y}{2}=55^o\)

\(\displaystyle x+x+y=180^o\)

Solving for x and y by substitution, we get x = 70° and y = 40° (which average out to 55°).

70 + 70 + 40 equals 180 also checks out.

Since 70° is not an answer choice for us, we know that the 40° must be one of the angles.

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}\(\displaystyle 30^{\circ}\). The measure of angle CBD is 60^{\circ}\(\displaystyle 60^{\circ}\). The length of segment \overline{AD}\(\displaystyle \overline{AD}\) is 4.

Find the measure of \dpi{100} \small \angle ADB\(\displaystyle \dpi{100} \small \angle ADB\).

Possible Answers:

30^{\circ}\(\displaystyle 30^{\circ}\)

90^{\circ}\(\displaystyle 90^{\circ}\)

15^{\circ}\(\displaystyle 15^{\circ}\)

60^{\circ}\(\displaystyle 60^{\circ}\)

45^{\circ}\(\displaystyle 45^{\circ}\)

Correct answer:

30^{\circ}\(\displaystyle 30^{\circ}\)

Explanation:

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

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

Example Question #2 : Triangles

The base angle of an isosceles triangle is 27^{\circ}\(\displaystyle 27^{\circ}\).  What is the vertex angle?

Possible Answers:

135^{\circ}\(\displaystyle 135^{\circ}\)

108^{\circ}\(\displaystyle 108^{\circ}\)

75^{\circ}\(\displaystyle 75^{\circ}\)

126^{\circ}\(\displaystyle 126^{\circ}\)

149^{\circ}\(\displaystyle 149^{\circ}\)

Correct answer:

126^{\circ}\(\displaystyle 126^{\circ}\)

Explanation:

Every triangle has 180 degrees.  An isosceles triangle has one vertex angle and two congruent base angles. 

Solve the equation 27+27+x=180\(\displaystyle 27+27+x=180\) for x to find the measure of the vertex angle. 

x = 180 - 27 - 27

x = 126

Therefore the measure of the vertex angle is 126^{\circ}\(\displaystyle 126^{\circ}\).

Example Question #62 : Triangles

An isosceles triangle has an area of 12. If the ratio of the base to the height is 3:2, what is the length of the two equal sides?

 

Possible Answers:

4√3

6

4

3√3

5

Correct answer:

5

Explanation:

Area of a triangle is ½ x base x height. Since base:height = 3:2, base = 1.5 height.  Area = 12 = ½ x 1.5 height x height or 24/1.5 = height2.  Height = 4.  Base = 1.5 height = 6. Half the base and the height form the legs of a right triangle, with an equal leg of the isosceles triangle as the hypotenuse. This is a 3-4-5 right triangle.

 Sat_math_167_01

 

 

 

 

Example Question #1 : How To Find The Length Of The Side Of An Acute / Obtuse Isosceles Triangle

Two sides of a triangle each have length 6. All of the following could be the length of the third side EXCEPT

Possible Answers:
11
2
12
3
1
Correct answer: 12
Explanation:

This question is about the Triangle Inequality, which states that in a triangle with two sides A and B, the third side must be greater than the absolute value of the difference between A and B and smaller than the sum of A and B.

Applying the Triangle Inequality to this problem, we see that the third side must be greater than the absolute value of the difference between the other two sides, which is |6-6|=0, and smaller than the sum of the two other sides, which is 6+6=12. The only answer choice that does not satisfy this range of possible values is 12 since the third side must be LESS than 12.

 

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

Find the area of a triangle whose base is \(\displaystyle 14in\) and whose height is \(\displaystyle 2ft\).

Possible Answers:

\(\displaystyle 136in^{2}\)

\(\displaystyle 336in ^{2}\)

\(\displaystyle 186in^{2}\)

\(\displaystyle 196in^{2}\)

\(\displaystyle 168in^{2}\)

Correct answer:

\(\displaystyle 168in^{2}\)

Explanation:

This problem is solved using the geometric formula for the area of a triangle.  

\(\displaystyle A=\frac{1}{2}b*h\)

Convert feet to inches.

\(\displaystyle \dpi{100} \frac{2ft}{1}*\frac{12in}{1ft}=24in\)

\(\displaystyle A=\frac{1}{2}*14in*24in\)

\(\displaystyle A=168in^{2}\)

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

If triangle ABC has vertices (0, 0), (6, 0), and (2, 3) in the xy-plane, what is the area of ABC?

Possible Answers:

10

9

12

18

20

Correct answer:

9

Explanation:

Sat-triangle

Sketching ABC in the xy-plane, as pictured here, we see that it has base 6 and height 3. Since the formula for the area of a triangle is 1/2 * base * height, the area of ABC is 1/2 * 6 * 3 = 9.

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

What is the area of a triangle with a height of \(\displaystyle 7\) and a base of \(\displaystyle 4\)?

Possible Answers:

\(\displaystyle 28\)

\(\displaystyle 14\)

\(\displaystyle 24\)

\(\displaystyle 12\)

Correct answer:

\(\displaystyle 14\)

Explanation:

When searching for the area of a triangle we are looking for the amount of the space enclosed by the triangle.

The equation for area of a triangle is

Plug the values for base and height into the equation yielding

 \(\displaystyle \frac{1}{2}(4)(7)=Area\)

Then multiply the numbers together to arrive at the answer \(\displaystyle 14\).

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