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QuestionFind the hypotenuse cc and angles AA and BB of a right triangle with sides a=78.2a=78.2 yd and b=40.2b=40.2 yd.

Studdy Solution

STEP 1

Assumptions1. We have a right triangle with sides a, b, and c. . Side a is the opposite side, b is the adjacent side, and c is the hypotenuse.
3. The angles of the triangle are A, B, and C (where C is the right angle).
4. The lengths of sides a and b are given as78.yd and40.yd respectively.
5. We need to find the length of side c and the measures of angles A and B.

STEP 2

We can find the length of side c using the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b).
c2=a2+b2c^2 = a^2 + b^2

STEP 3

Substitute the given values of a and b into the equation.
c2=(78.2 yd)2+(40.2 yd)2c^2 = (78.2\,yd)^2 + (40.2\,yd)^2

STEP 4

Calculate the squares of a and b.
c2=6112.84 yd2+1616.04 yd2c^2 =6112.84\,yd^2 +1616.04\,yd^2

STEP 5

Add the squares of a and b.
c2=7728.88 yd2c^2 =7728.88\,yd^2

STEP 6

Take the square root of both sides to solve for c.
c=7728.88 yd2c = \sqrt{7728.88\,yd^2}

STEP 7

Calculate the square root.
c=87.9 ydc =87.9\,yd

STEP 8

Now we can find the measures of angles A and B using the trigonometric functions sine, cosine, and tangent. Let's start with angle A. We can use the tangent function, which is the ratio of the opposite side to the adjacent side.
tan⁡(A)=ab\tan(A) = \frac{a}{b}

STEP 9

Substitute the given values of a and b into the equation.
tan⁡(A)=78.2 yd40.2 yd\tan(A) = \frac{78.2\,yd}{40.2\,yd}

STEP 10

Calculate the tangent of A.
tan⁡(A)=.944\tan(A) =.944

STEP 11

To find the measure of angle A, take the arctangent of the calculated value. Remember to convert the result from radians to degrees.
A=arctan⁡(.944)A = \arctan(.944)

STEP 12

Calculate the arctangent.
A=62.6∘A =62.6^\circ

STEP 13

Now let's find the measure of angle B. We know that the sum of the measures of the angles in a triangle is180 degrees, and since one of the angles is a right angle (90 degrees), we can find B by subtracting the measures of angles A and C from180.
B=180∘−A−CB =180^\circ - A - C

STEP 14

Substitute the values of A and C into the equation.
B=180∘−62.6∘−90∘B =180^\circ -62.6^\circ -90^\circ

STEP 15

Calculate the measure of angle B.
B=27.4∘B =27.4^\circThe solution to the problem is c =87.9 yd, A =62. degrees, B =27.4 degrees.

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