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Calculating with triangles

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The base angles of an isosceles triangle are equal.

If a = b, then x = y

If the base angles of a triangle are equal, the triangle is isosceles.

If x = y, then a = b

The measure of an exterior angle is equal to the sum of the measures of the remote interior angles.

l is a straight line.

Then, x = y + z

In a triangle, the greater angle lies opposite the greater side.

If a < b, then y < x

If a < b, then y < x

Similar Triangles

If ΔABC ~ ΔDEF, then

m∠A = m∠B

m∠B = m∠E

m∠C = m∠F

and

The sum of the interior angles of a triangle is 180 degrees.

m∠A + m∠B + m∠A  = 180º

The area of a triangle is one-half the product of the altitude to a side and the side.

Area of △ABC =

Note: If m∠A = 90º

Area also =

In a right triangle

c2 = a2 + b2

and x°+ y° = 90°

Memorize the following standard triangles

The sum of the lengths of two sides of a triangle is greater than the length of the third side. (This is like saying that the shortest distance between two points is a straight line.)

a + b > c

a + c > b

b + c > a

Example 1

In the diagram below, what is the value of x?

(A) 20

(B) 25

(C) 26

(D) 45

(E) 48

Solution

Choice C is correct.

Method 1: Use right triangle. Then,

x2 = 242 + 102

= 576 + 100

= 676

Thus, x = 26 (Answer)

Method 2: Notice that ΔMNP is similar to one of the standard triangles:

This is true because

12/24 = 5/10

Hence, 12/24 =13/x or x = 26 (Answer)

Example 2

If Masonville is 50 kilometers due north of Adamston and Elvira is 120 kilometers due east of Adamston, then the minimum distance between Masonville and Elvira is

(A) 125 kilometers

(B) 130 kilometers

(C) 145 kilometers

(D) 160 kilometers

(E) 170 kilometers

Solution

Choice B is correct. Draw a diagram first.

The given information translates into the diagram above.

The triangle above is a multiple of the special 5–12–13 right triangle.

50 = 10×5

120 = 10×12

Thus, x = 10×13 = 130 kilometers

(Note: The Pythagorean Theorem could also have been used: 502 + 1202 = x2.)

Example 3

In triangle ABC, if a > c, which of the following is true?

(A) BC = AC

(B) AB > BC

(C) AC > AB

(D) BC > AB

(E) BC >AC

Solution

Choice D is correct.

From basic geometry, we know that, since m∠BAC > m∠BCA, then leg opposite BAC > leg opposite BCA, or BC > AB

Example 4

The triangle above has side BC = 10, angle B = 45°, and angle A = 90°. The area of the triangle

(A) is 15

(B) is 20

(C) is 25

(D) is 30

(E) Cannot be determined.

Solution

Choice C is correct.

First find angle C.

90° + 45° + m∠C = 180°

So m∠C = 45°.

We find AB = AC, since m∠B = m∠C = 45°.

Since our right triangle ABC has BC = 10, (the right triangle ) multiply by 10 to get a right triangle:

Thus side AB =

side AC =

Now the area of triangle ABC is

Example 5

In the figure above, what is the value of x?

(A) 30

(B) 40

(C) 50

(D) 80

(E) 100

Solution

Choice B is correct.

Remember triangle facts. Use Statement II.

ADB is an exterior angle of ΔACD, so mADB = x + x = 2x          (1)

In ΔADB, the sum of its angles = 180, so

mADB + 55 + 45 = 180

or   mADB + 100 = 180

or             mADB = 80                                                                   (2)

Equating (1) and (2) we have 2x = 80, x = 40 (Answer)

Example 6

Which of the following represents all of the possibilities for the value of a in the figure above?

(A) 1 < a < 9

(B) 4 < a < 5

(C) 0 < a < 9

(D) 4 < a < 9

(E) 5 < a < 9

Solution

Choice A is correct. Since the sum of the lengths of two sides of a triangle is greater than the length of the third side, we have:

a + 5 > 4      (1)

a + 4 > 5      (2)

5 + 4 > a      (3)

From (2) we get: a > 1.

From (3) we get: 9 > a.

This means that 9 > a > 1, or 1 < a < 9.

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