Solve each problem. See Examples 1–4. Altitude of a Triangle Fin...

Question

Solve each problem. See Examples 1–4.  Altitude of a Triangle Find the altitude of an isosceles triangle having base 184.2 cm if the angle opposite the base is 68°44'.

Accepted Answer

Hello, everyone. We are given that Isosceles triangle has a base of 232.6 centimeters. We want to calculate its altitude. If the angle opposite the base has a measure of 51 degrees 16 minutes, we want a round, our answer to one decimal place. Our answer choices are a 261. centimeters. B 237.1 centimeters C 242.4 centimeters D 256.9 centimeters. So first, I'm gonna draw a nice sale triangle. Mccole ISOS triangle has a base where both angles are congruent and the sides opposite that base are also congruent. We're told that the base is, has a length of 232.6 centimeters. We want to calculate the length of its altitude. So we called an altitude starts at a vertex. And in NYO triangle, it is a perpendicular bisector. So this will make a 90 degree angle and both segments will be equal to each other. And we know the angle that is opposite. The base has a measure of 51 degrees 16 minutes. So that's gonna be the angle at the top in my picture and that is 51 degrees 16 minutes. All right. So what we ultimately want to find is the length of this altitude. So I'm gonna call that X I think to make this easier, I want to find what is the measure of the base angles. So recalling that the triangles interior angles total to 100 80 degrees. I know that if I call both of these unknown angles, let's call them Y for now because they both measure the same, it'd be two Y plus 51 degrees. 16 minutes equals 100 80 degrees. So solving for Y I'm gonna subtract 51 degrees, 16 minutes from both sides. And when we subtract, we get 128 degrees, 44 minutes. So two Y equals 128 degrees. 44 minutes divide both sides by two. And we find that each angle in our base is 64 degrees 22 minutes. I want to convert that to a decimal to make it a little easier to use in my calculator when I go to calculate my X. So I'm gonna take the 22 minutes, divide it by 60 then add it to the 64 degrees and we get 64. degrees. So that's our angle at our base 64. degrees. So now still looking at my base measure of 232.6 centimeters. If I'm only gonna use half of that, I should divide this by two. That way, I know what the measure I'm working with is. So if I do 232.6 centimeters divided by two, we get 116.3 centimeters. So this means that half of it. So this one side is 116.3 centimeters. So now looking at what I have versus what I need to solve, I have an angle of 64.36667 degrees. I have the side adjacent to it and I want to find the side opposite it. So opposite and adjacent leads me to tangent. So we're gonna do the tangent of 64.36667 degrees equals our opposite side X divided by our adjacent side, 116.3 to get the X by itself, I multiply both sides by 116.3. So in my calculator, I'm gonna enter 116.3 multiplied by the tangent of 64. degrees. Make sure your calculator is in degree mode because otherwise you will get the wrong answer and I get 242. nine. and this would be centimeters. We want this rounded to one decimal place. So I'm looking at the threes in that one decimal place the number to the right of it is greater than five. So I'm gonna round up. So our altitude, which I called X is gonna be 242.4 centimeters. And that is answer choice C have a nice day.

Solve each problem. See Examples 1–4. Altitude of a Triangle Find the altitude of an isosceles triangle having base 184.2 cm if the angle opposite the base is 68°44'.

Key Concepts

Isosceles Triangle Properties

Trigonometric Ratios

Angle Conversion

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