![]() In addition, another important property to know is that the length of each leg of an isosceles. So in a sense you don't even need to find the legs: in an isosceles right triangle, the hypotenuse uniquely determines the legs, and vice versa. This can be represented using the following equation: c a x b. In that light we could make this even shorter by noting: Since the triangle is isosceles and right, the legs are equal ( $a=b$) and are given by $h/\sqrt 2$. To actually further this discussion and extend to isosceles right triangles, suppose you have only the hypotenuse $h$. In right triangles, the legs can be used as the height and the base. Where $a,b$ are the legs of the triangle. That the question specifies this also may be indicative that your "shortcut" was the intended method (though kudos to you for finding an additional method either way!).Īs is probably obvious whenever you draw right triangles, its area can be given by That is our area.After the edit to the OP, yeah, as pointed out by Deepak in the comments: it is because the triangle is not just any isosceles triangle, but an isosceles right triangle. Well, that's just going to be equal to one half times 10 is five, times 12 is 60, 60 square units, whatever Calculate area, perimeter of an isosceles triangle step-by-step. Finally, solve the equation to find the unknown base, x. Then, use the Pythagorean theorem to create an equation involving x. So, our base is that distance which is 10, and now we know our height. To find the value of a base (x) in an isosceles triangle, first split the triangle into two congruent right triangles by drawing an altitude. Well, we already figured out that our base is this 10 right over here, let me do this in another color. ![]() Remember, they don't want us to just figure out the height here, they want us to figure out the area. Purely mathematically, you say, oh h could be plus or minus 12, but we're dealing with the distance, so we'll focus on the positive. And what are we left with? We are left with h squared is equal to these canceled out, 169 minus 25 is 144. We can subtract 25 from both sides to isolate the h squared. ![]() To be equal to 13 squared, is going to be equal to our longest side, our hypotenuse squared. H squared plus five squared, plus five squared is going Pythagorean Theorem tells us that h squared plus five The Pythagorean Theorem to figure out the length of Two congruent triangles, then we're going to split this 10 in half because this is going to be equal to that and they add up to 10. I was a little bit more rigorous here, where I said these are How was I able to deduce that? You might just say, oh thatįeels intuitively right. So, this is going to be five,Īnd this is going to be five. Going to have a side length that's half of this 10. That is if we recognize that these are congruent triangles, notice that they both have a side 13, they both have a side, whatever And so, if you have two triangles, and this might be obviousĪlready to you intuitively, where look, I have two angles in common and the side in between them is common, it's the same length, well that means that these are going to be congruent triangles. So, that is going to be congruent to that. And so, if we have two triangles where two of the angles are the same, we know that the third angle Point, that's the height, we know that this is, theseĪre going to be right angles. And so, and if we drop anĪltitude right over here which is the whole And so, these base angles areĪlso going to be congruent. ![]() It's useful to recognize that this is an isosceles triangle. But how do we figure out this height? Well, this is where One half times the base 10 times the height is. So, if we can figure that out, then we can calculate what ![]() But what is our height? Our height would be, let me do this in another color, our height would be the length Our base right over here is, our base is 10. That the area of a triangle is equal to one half times Recognize, this is an isosceles triangle, and another hint is that And see if you can find the area of this triangle, and I'll give you two hints. ![]()
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