Theory and Examples

The acco...

Question

Theory and Examples

The accompanying figure shows a rectangle inscribed in an isosceles right triangle whose hypotenuse is 2 units long.

a. Express the y-coordinate of P in terms of x. (You might start by writing an equation for the line AB.)

Accepted Answer

Welcome back everyone to another video. The diagram displays a rectangle drawn inside an isosceles right triangle with a hypotenuse that is 18 units long. Write the Y coordinate of point P as a function of X. Where given for an intraces ASY equals 9 x + 9 B X. 9 C X + 9, and D-9 X + 1. So let's analyze the given triangle. Let's let's add a point C at a coordinate -90 and let's add a.0 which represents the origin, right? Essentially, as we understand, this is an isosceles right triangle with a hypotenuse of 18 units. So that would be our AC. We can say that length AC is equal to 18. We can see that because we're. Acting 9 minus -9, this gives us 18 units. Essentially what we can do is analyze the height of OB, right? We are going to identify the length of OB and we can do this very simply. We understand that our axes intersect at the right angle, so we have the height of that triangle going from the vertex going down, right? So essentially for an isosceles triangle, if it is a right triangle, we're going to label our angle of 45 degrees at the base, right? Because we have two equal side lengths. Let's understand that LBC equals L of AB. Those sidelines are the same. We can call them A and A, right? So if they are equal to each other, we will have 180 minus 90 divided by 2. This gives us 45 degrees at the base. And now, because we have the right angle at the base, we have to understand that the remaining angle, which is CBO is going to be 45 degrees, and then ABO is going to be 45 degrees as well. So now we have two. Additional isosceles right triangles that we can analyze, but, but most importantly, we can now determine that BO is also 9 units in length, right, because we can see that OC and OA, those are 9 units in length, they're just half of 18. So now length of OB is simply equal to the length of OC, which is one half of the length of AC, that's the least we have so far analyzed, and one half of AC is simply 9 units that we, we know our height. And we can label the coordinates of point B. So point B has the coordinates of X of 0 and the Y coordinate is going to be 9, right, because we're going from 0 up to 9 to get that length of 9. So the coordinates of B are 0.9, then the coordinates of point A are 9.0, and now we can analyze. The equation of the line passing through the points B and A, right? First of all, we're going to identify the slope. Let's suppose that our line has an equation of M X + B using the equation of a slope, we know that we're evaluating the difference in Y. Divided by the difference in X, so we can take two points, for example, Y of point B minus Y of point A, and divide by X of point B minus X of point A. So what is Y of B? Well, it's 9. Y A is 0, so we subtract 9 minus 0 and divide by X of B, which is 0 minus X of A, which is 9. So we get 9 divided by -9 and that's -1. So, our slope is -1, and now we can use the Win slope formula, right? Essentially we're taking Y minus Y1 is equal to M multiplied by X minus X1. In this case, we can take any point that we want. Let's suppose that we're taking. A, so Y minus Y A equals m multiplied by x minus X of A. So we get Y minus what is Y A while it's 0 equals our slope -1 multiplied by X minus X of A. So that's 9. And now solving for Y we get Y equals negative X plus 9. So we have expressed the equation of the line and that line contains. The point B, right, because that point B is on that line. So if we know the coordinate of P X, right, then we are going to identify the Y coordinate of the same point P. Well then, so we have identified the required equation, as we can see the correct answer to this problem corresponds to the answer choice C. Thank you for watching.

Theory and Examples

The accompanying figure shows a rectangle inscribed in an isosceles right triangle whose hypotenuse is 2 units long.

a. Express the y-coordinate of P in terms of x. (You might start by writing an equation for the line AB.)

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Key Concepts

Isosceles Right Triangle Properties

Equation of a Line

Inscribed Figures

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