# Two right triangles with shared hypotenuse

#### WillB

I have two RA triangles with a shared common hypotenuse. Given the lengths of the opposite sides of the two right angle triangles, a1 and a2, and the sum of both angles, theta, how can I calculate each angle, theta1 and theta2?
Thanks.

#### skipjack

Forum Staff
What have you tried? Do you have specific values for a1, a2 and theta?

#### WillB

For example:
a1 = 3.5 inches, a2 = 1.5", theta = 45 degrees.
I've drawn it graphically but would like to develop a spreadsheet since those variables will change in my work.
Thanks.

Hypotenuse = a1/sinØ1 = a2/sinØ2, but not sure where to go from there.

And sinØ2 = sin(Ø-Ø1)

#### Yooklid

Extend the vertical side of the bottom triangle so that it intersects the upper line. This will form a 3rd triangle with two sides and an angle known. From this you can calculate all remaining angles and sides.

Thanks Yooklid

#### skipjack

Forum Staff
. . . with two sides and an angle known.
I don't think two sides are known.

Let's use θ instead of Ø, and θ-β and β for the two unknown angles.

Let the known sides be denoted by a and b, with a = 3.5 opposite θ-β and b = 1.5 opposite β.
For θ = 45°, cos(θ) = sin(θ) = 1/√2.

As hypotenuse = a/sin(θ-β) = b/sin(β), a*sin(β) = b*sin(θ-β) = b(sin(θ)cos(β) - cos(θ)sin(β)),
and so tan(β) = sin(β)/cos(β) = b(sin(θ)/(a + b*cos(θ)).

In your case, tan(β) = (1.5/√2)/(3.5 + 1.5/√2) = 3/(3 + 7√2), so β = 13.09° approximately.

#### Yooklid

I don't think two sides are known.
Yes, my statement was incorrect. However one side is known as well as all of the angles. So the solution can still proceed from there.

Using the notation in the attachment (I made this before seeing Skipjack's notation, so I used A, B and C for the angles):
Angles of triangle PQT are A, 90, 90-A.
Therefore, angles of RST are also A, 90, 90-A.
Side RS is known. So all other sides of RST can be determined.
Now, all sides of PQT can be determined.
Triangle sides PS and RS are known, so common hypotenuse PR can be determined.
Angles B and C can now be determined.

#### Attachments

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#### skipjack

Forum Staff
Finding PQ suffices, as tan(C) = RQ/PQ can then be found. What you get is equivalent to the formula I gave.

It works:

Thanks Skipjack!