Isosceles Triangle side lengths?

Question

I need to be able to figure out the two congruent side lengths of an isosceles triangle. The two figures I will know is the base and the vertex angle. The base is always going to be 1.75. But the vertex angle can change. Its easy to find out the two congruent angles by simply taking 180 - vertex angle and divide by 2. So I now have all three angles, and the base lenght of 1.75.

Now how do I find the lengths of the two congruent sides (I forgot the math since I have been out of school for some many years). If it makes it easier you can cut the Isosceles into 2 right triangles. If you break it into two right triangles I will need to know the math to get the hypotenuse of the right triangle. If it helps you can use the following numbers for the isosceles... Vertex angle = 80, congruent angles = 50, and base is 1.75".

Thanks in advance!!!

but to do Pythagora: a^2 +b^2 =c^2 where c is the hypothenuse, I would need to know at least 2 sides, and I dont, not at least of the right triangle. Of the isosceles triangle at least I know the two sides are equal.

Answer 1

Each congruent side will act as a hypotenuse to the two right triangles formed by one-half the base and the height.

If all you're looking for is the length of the congruent sides, use the cosine function on your calculator. The cosine of an angle of a right triangle is equal to the ratio of the adjacent leg (half the base) to the hypotenuse (each congruent side).

cos(Base Angle) = (half the base) / (either congruent side)

A little manipulation gives

(either congruent side) = (half the base) / cos(Base Angle)

In your example, half the base = 1.75" / 2 = 0.875".
If the base angle measures 50°, then the congruent sides are each
0.875" / cos(50°) = approx. 0.875" / 0.6427876, or
about 1.36 inches.

As another example, if your base angle measures 70°, then the congruent sides are each
0.875" / cos(70°) = approx. 2.56 inches.

Answer 2

In an isosceles right triangle the two equal sides must be the legs, since the hypotenuse is always the longest side in a right triangle. Let x be the length of a leg. Then, by Pythagoras, the length of the hypotenuse is: c = √(x² + x²) = √(2x²) = (√2) x But, you're told that c is 4 cm. longer than x: c = x + 4 (√2) x = x + 4 ... substituted for c (√2)x - x = 4 ... add -x to both sides (√2 - 1)x = 4 .... factor on the left x = 4/(√2 - 1) .... divide both sides by (√2 - 1) So, the answer is 4/(√2 - 1) centimeters.

Answer 3

This is trigonometry.

An isosceles triangle is bisected by its altitude into two congruent right triangles. If you are given the vertex angle v, do the following:

Determine sin (v/2). In the example you provided, sin (80/2) = .6428

This is a ratio of the opposite leg divided by the hypotenuse. Since you know the opposite leg is 1.75/2 = 0.875 inches, the math is simple.

.6428 = .875 / h
.6428h = .875
h = 1.361 in

Answer 4

Once you divide that triangle into two right triangles, you only know the length of one side and that is half of that 1.75 which is .875. you will either need to use the cos 50 = .875/x or sin 40 = .875/x to get the value of x which is the length of the two congruent sides. It's calculator work after this...

Answer 5

a^2 + b^2 = c^2 for the hypotenuse. You'll need the height to the vertex by the 50% angle at the base, measure that, and you can find the hypotenuse. That's all I remember short of pulling out the geometry & trig books...

Answer 6

the easiest way is to break it up into the two right triangles and use trig

the cosine of the angle will equal the adjacent length over the hypoteneus length, or the hypoteneuse will equal adjacent divided by the cosine of the angle

in this case that means that the cosine of the angle will equal the hypoteneus (the length in question) dvivded by half of 1.75

so, if you know the angle, have a cheap calculator with trig functions, take the cosine of the angle, and divide the your base (half your original base) by this cosine value, and that is your hypoteneuse

Answer 7

Here is my solution:

http://www.geocities.com/maxpaynecocktail/answer-isosceles.jpg

The answer to the question is:

Let theta = t
Then, x = 7/[8sin(t/2)]

This will work for any given vertex angle with the given base.

Answer 8

Here Trigonometry will come to help. ie, (1.75/2)csc(theta)= Hypotenuse which is one of the equal sides of the isosceles triangle.
For example when theta is very near to 90 degree or as theta approaches 90 degree (1.75/2)csc(theta)= (1.75/2) ( or approximately equal to (1.75/2))
when theta is 45degree it is =(1.75/sqrt2)
Given theta is 80 degree we have (1.75/2)csc(80)=(1.75/2)* 1.015

Answer 9

Asquared + Bsquared = Csquared.....

so when you divide the triangle you have two right triangles. So you formula is pretty much uses:

Asquared = 1/2 base = .875
Bquared =

oops...i got stuck.....sorry man...thought I had it. haha...gotta get out of this office now. good luck

Answer 10

um i would say you should use side angle side to find the answer but i forgot the formula and i don't know how to put it on here but type in SAS side angle side for Math B and they will give it to you in goole or answers.com