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Altitudes AD and BE of acute triangle ABC intersect at point H. If angle ABH = 30 and angle DAC = 45, then what is angle HCA in degrees?

 

 Jun 2, 2024

Best Answer 

 #1
avatar+806 
+1

First, let's note somethings. 

Notice that right triangles ABC, ABE, ABD, BHD, BEC are 30-60-90 triangles.

Right triangles ABe and ADC are 45-45-90 triangles. 

 

Now, let's draw the altitude for triangle ABC by connecting point C to line segment AB.

This line intersects point H on the way. 

 

We know that BAC is is 60 degrees, so that leaves HCA would be 30 degrees. 

 

So 30 degrees is our answer!

 

Thanks! :)

 Jun 2, 2024
 #1
avatar+806 
+1
Best Answer

First, let's note somethings. 

Notice that right triangles ABC, ABE, ABD, BHD, BEC are 30-60-90 triangles.

Right triangles ABe and ADC are 45-45-90 triangles. 

 

Now, let's draw the altitude for triangle ABC by connecting point C to line segment AB.

This line intersects point H on the way. 

 

We know that BAC is is 60 degrees, so that leaves HCA would be 30 degrees. 

 

So 30 degrees is our answer!

 

Thanks! :)

NotThatSmart Jun 2, 2024

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