# Question about a simple triangle property **URL:** \ **Category:** Factual Questions\ **Created:** [September 11, 2007, 7:02pm UTC](https://boards.straightdope.com/t/question-about-a-simple-triangle-property/418815 "2007-09-11T19:02:44Z")\ **Posts on this page:** 7\ **Page:** 1
**Author:** ![Earl\_Snake-Hips\_Tucker](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/earl_snake-hips_tucker/32/3269_2.png) [@Earl\_Snake-Hips\_Tucker](https://boards.straightdope.com/u/Earl_Snake-Hips_Tucker)\ **Post date:** [September 11, 2007, 7:02pm UTC](https://boards.straightdope.com/t/question-about-a-simple-triangle-property/418815/1 "2007-09-11T19:02:44Z")
And, no, it definitely ain’t homework. Let’s say I have a garden variety right triangle with sides a, b, and c, with c being the hypotenuse. Currently, it’s sitting flat on one if it sides with the right angle on the left, and the hypotenuse sloping downward to the right. I take the same triangle, and set it such that it is lying on its hypotenuse. It has a height h formed by dropping a line from the angle between a and b to the hypotenuse. The area can be determined by either ½a_b or ½c_h. So, the product of the two short sides equals the hypotenuse times the height. The GQ: What’s the easiest way of showing that in a right triangle that a_b = c_h? I know you can grind it out formulaicly, but what’s the slickest way? ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [September 11, 2007, 7:14pm UTC](https://boards.straightdope.com/t/question-about-a-simple-triangle-property/418815/2 "2007-09-11T19:14:57Z")
> [@](#): > > What’s the easiest way of showing that in a right triangle that a_b = c_h? I know you can grind it out formulaicly, but what’s the slickest way? You just did it. Both give the area, and the area can’t change, so they must be the same. ---
**Author:** ![MikeS](https://avatars.discourse-cdn.com/v4/letter/m/919ad9/32.png) [@MikeS](https://boards.straightdope.com/u/MikeS)\ **Post date:** [September 11, 2007, 7:15pm UTC](https://boards.straightdope.com/t/question-about-a-simple-triangle-property/418815/3 "2007-09-11T19:15:04Z")
Similar triangles. Let ABC be the triangle, with the right angle at C. The triangles ABC and CBD share the angle at B, and so the ratios of their sides are the same. In particular, BC/CD = AC/AB, or in your language b/c = a/h. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [September 11, 2007, 8:03pm UTC](https://boards.straightdope.com/t/question-about-a-simple-triangle-property/418815/4 "2007-09-11T20:03:02Z")
Psst, **MikeS** , that should be b/c = h/a. ---
**Author:** ![Quercus](https://avatars.discourse-cdn.com/v4/letter/q/7ab992/32.png) [@Quercus](https://boards.straightdope.com/u/Quercus)\ **Post date:** [September 11, 2007, 8:15pm UTC](https://boards.straightdope.com/t/question-about-a-simple-triangle-property/418815/5 "2007-09-11T20:15:34Z")
I think **MikeS** has the concept (if you want to omit the equal area proof), but with a typo. In case it’s not clear, D is the point on the original hypotenuse (AB) which forms the height (right-angle line from C to AB). And A is the point opposite segment a, etc. Then his last line should be BC/CD=AB/AC, or in your language a/h = c/b, so a_b = c_h (On preview, what **Chronos** said) ---
**Author:** ![MikeS](https://avatars.discourse-cdn.com/v4/letter/m/919ad9/32.png) [@MikeS](https://boards.straightdope.com/u/MikeS)\ **Post date:** [September 11, 2007, 8:38pm UTC](https://boards.straightdope.com/t/question-about-a-simple-triangle-property/418815/6 "2007-09-11T20:38:58Z")
That’ll teach me to dash off a quick post without carefully checking it. Yes, the point D is the intersection of the hypotenuse and its perpendicular line passing through C; and my equation should in fact read either b/c = h/a or c/b = a/h. ---
**Author:** ![Joey\_P](https://avatars.discourse-cdn.com/v4/letter/j/919ad9/32.png) [@Joey\_P](https://boards.straightdope.com/u/Joey_P)\ **Post date:** [September 12, 2007, 3:29am UTC](https://boards.straightdope.com/t/question-about-a-simple-triangle-property/418815/7 "2007-09-12T03:29:48Z")
[QUOTE=Earl Snake-Hips Tucker] The area can be determined by either **½a_b or ½c_h**. So, the product of the two short sides equals the hypotenuse times the height. The GQ: What’s the easiest way of showing that in a right triangle that a_b = c_h? I know you can grind it out formulaicly, but what’s the slickest way? [/QUOTE] ½a_b=area AND area= ½c_h So we have ½a_b = ½c_h Multiply each side by 2 and you’re done.