So the key of realization here is isosceles triangle, the altitudes splits it into
ISOSCELES TRIANGLE FORMULA ANGLES PLUS
So this length right over here, that's going to be five and indeed, five squared plus 12 squared, that's 25 plus 144 is 169, 13 squared. This distance right here, the whole thing, the whole thing is So x is equal to the principle root of 100 which is equal to positive 10. But since we're dealing with distances, we know that we want the This purely mathematically and say, x could be
Is equal to 25 times four is equal to 100. We can multiply both sides by four to isolate the x squared. So subtracting 144 from both sides and what do we get? On the left hand side, we have x squared over four is equal to 169 minus 144. That's just x squared over two squared plus 144 144 is equal to 13 squared is 169. This is just the Pythagorean Theorem now. We can write that x over two squared plus the other side plus 12 squared is going to be equal to We can say that x over two squared that's the base right over here this side right over here. Let's use the Pythagorean Theorem on this right triangle on the right hand side. And so now we can use that information and the fact and the Pythagorean Theorem to solve for x. So this is going to be x over two and this is going to be x over two. So they're both going to have 13 they're going to have one side that's 13, one side that is 12 and so this and this side are going to be the same. And since you have twoĪngles that are the same and you have a side between them that is the same this altitude of 12 is on both triangles, we know that both of these So that is going to be the same as that right over there. Because it's an isosceles triangle, this 90 degrees is the Is an isosceles triangle, we're going to have twoĪngles that are the same. Well the key realization to solve this is to realize that thisĪltitude that they dropped, this is going to form a right angle here and a right angle here and notice, both of these triangles, because this whole thing The two key facts we used in this question were, firstly, that the angle sum in any triangle is 180 degrees and, secondly, that in an isosceles triangle the two base angles are equal.To find the value of x in the isosceles triangle shown below. To find the value of 𝑦, we need to divide both sides by nine. This will give nine 𝑦 is equal to 41 plus four. To do this, I’m going to choose to substitute 𝑥 equals 41 into equation two. The final step is to divide both sides of the equation by two. Next, we subtract four from both sides, giving two 𝑥 is equal to 82. Combining the like terms, the two 𝑥s gives two 𝑥 plus four is equal to 86. So substituting 𝑥 plus four in place of nine 𝑦 in the first equation gives 𝑥 plus four plus 𝑥 is equal to 86. And therefore, the most straightforward method of solution is going to be to substitute the expression for nine 𝑦 from the second equation into the first equation. In order to find the values of 𝑥 and 𝑦, we need to solve these two equations simultaneously. And the second: nine 𝑦 is equal to 𝑥 plus four.
The first equation: nine 𝑦 plus 𝑥 equals 86. So now we have two equations with two unknowns. Adding three to both sides of this equation simplifies it slightly, to give nine 𝑦 is equal to 𝑥 plus four. Nine 𝑦 minus three is equal to 𝑥 plus one. Therefore, we can form a second equation involving the measures of these two angles. And in terms of the angles, it means that the two base angles, those currently shaded in orange, must be equal to each other. This means that triangle 𝐴𝐵𝐶 is an isosceles triangle. We’re told in the diagram that two of the sides of this triangle are of the same length, 𝐴𝐵 and 𝐴𝐶. Let’s consider what else we know about this triangle. We need another equation in order to be able to find the values of 𝑥 and 𝑦. But we aren’t yet in a position to do so as we have just one equation with two unknowns. Now we want to calculate the values of 𝑥 and 𝑦. Subtracting 94 from both sides of the equation simplifies it further, giving nine 𝑦 plus 𝑥 is equal to 86. We therefore have nine 𝑦 plus 𝑥 plus 94 is equal to 180. On the left-hand side, we have negative three plus one plus 96. This equation can be simplified slightly. Nine 𝑦 minus three plus 𝑥 plus one plus 96 is equal to 180. We can therefore form an equation involving the sizes of the three angles. The first fact that we know about the angles in any triangle is their sum is 180 degrees. In order to do this, we’ll need to solve some equations. And the other angles are expressed in terms of these variables, 𝑥 and 𝑦, whose values we wish to calculate. So we have a diagram of a triangle 𝐴𝐵𝐶 in which we’re told one of the angles is 96 degrees.
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