DS 700 level question

[kishoretvk]

If vertices of a triangle have coordinates (-2,2), (3,2) and (x,y), what is the area of the triangle?

A) |y-2|=1
B) angle at the vertex (x,y) equals to 90 degrees

QA is quiet weird so, i've posted here for expert explanation


QA is A :(


my approach :
s1: this gives two values for Y i.e. 3 and -1 so its insufficient as we dont know any info other than this we need to know either value or x or the nature of triange.
s2: now given angle is 90, here if the angle is 90 then we get a equation from slope, two variables and one equation so again NS

now consider s1 and s2, we have y two values and each of which will yield 2 different values for x : for example

use slope form y2-y1/ x2-x1 => u get equation as x2 - x-5= 0
so again NS
please let me know where did i go wrong

[RonPurewal]

the two points (-2, 2) and (3, 2) are spaced horizontally from each other. so, you can use that horizontal side as the "base" of the triangle.
since the base is thus a fixed number (and area is 1/2 x B x H), all we need is the height of the triangle.

because the triangle's designated base is horizontal, this "height" is vertical. so, it's just the difference between y = 2 and the y-coordinate of the third vertex.
(if you don't see why this is the case, draw a picture -- put the points (-2, 2) and (3, 2) on there, and consider a bunch of different triangles containing that side as a base.)

as you wrote, statement 1 gives either y = 1 or y = 3.

* if y = 1, then the height of the triangle is 2 - 1 = 1.

* if y = 3, then the height of the triangle is 3 - 2 = 1.

in either case, the height is the same. so, we've got a fixed base and a fixed height, so the area is a unique value. suficiente.

[milinjc]

Hey,

Thanks for the explanation.

I have one question about the Statement II.

B) angle at the vertex (x,y) equals to 90 degrees

Statement two translates, the triangle is right triangle. So, the line segment containing (-2,2), (3,2) vertices becomes a hypotenuse with value 5 and since there could only be one right triangle with 5 as hypotenuse value (5-4-3), statement II is sufficient to calculate the area of the triangle.

Milin

[RonPurewal]

Hey,

Thanks for the explanation.

I have one question about the Statement II.

B) angle at the vertex (x,y) equals to 90 degrees

Statement two translates, the triangle is right triangle. So, the line segment containing (-2,2), (3,2) vertices becomes a hypotenuse with value 5 and since there could only be one right triangle with 5 as hypotenuse value (5-4-3), statement II is sufficient to calculate the area of the triangle.

Milin

And why couldn't it be one of the other two sides?

[DDK]

Hey,

Thanks for the explanation.

I have one question about the Statement II.

B) angle at the vertex (x,y) equals to 90 degrees

Statement two translates, the triangle is right triangle. So, the line segment containing (-2,2), (3,2) vertices becomes a hypotenuse with value 5 and since there could only be one right triangle with 5 as hypotenuse value (5-4-3), statement II is sufficient to calculate the area of the triangle.

Milin

And why couldn't it be one of the other two sides?

The horizontal between (-2,2) and (3,2) must be the hypotenuse because it is opposite the right angle. However, just because the hypotenuse has a length of 5 does not mean the triangle is a 3-4-5 right triangle.

The two short sides of the triangle (adjacent to the right angle) can have any lengths such that L1^2 + L2^2 = 5^2, where L1 and L2 are the lengths of the two short sides. That is why statement 2 is insufficient.

[jnelson0612]

Hey,

Thanks for the explanation.

I have one question about the Statement II.

B) angle at the vertex (x,y) equals to 90 degrees

Statement two translates, the triangle is right triangle. So, the line segment containing (-2,2), (3,2) vertices becomes a hypotenuse with value 5 and since there could only be one right triangle with 5 as hypotenuse value (5-4-3), statement II is sufficient to calculate the area of the triangle.

Milin

And why couldn't it be one of the other two sides?

The horizontal between (-2,2) and (3,2) must be the hypotenuse because it is opposite the right angle. However, just because the hypotenuse has a length of 5 does not mean the triangle is a 3-4-5 right triangle.

The two short sides of the triangle (adjacent to the right angle) can have any lengths such that L1^2 + L2^2 = 5^2, where L1 and L2 are the lengths of the two short sides. That is why statement 2 is insufficient.

Nice work, DDK!