Grades 9-10 Video Solutions 2021-video 2021 9-10/9

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Problem number nine states, Ali drew three triangles on a grid. Exactly two of them have the same area. Exactly two of them are isosceles, and exactly two are right triangles.  Two of the triangles are shown. Which could be the third? So we know that Ali drew three triangles. Two of them were isosceles, two were right, and two had the same area.  Let's figure out which of these properties the two triangles that we have contain so we can see what the properties of the third one are.  First let's consider if the triangles are isosceles. Isosceles means that two of the three side lengths are equal to one another.  The first triangle has side lengths four and five, and a hypotenuse which comes out as a decimal. And the second triangle has side lengths four and four, and a hypotenuse which also  comes out as a decimal. So only the second one is isosceles in this case. Since Ali drew two isosceles triangles, we know that the third one must be isosceles.  Next let's see about right triangles. Both triangles one and two are right, meaning that the third triangle cannot be right. Now let's consider the same area.  To find the area of a triangle, we multiply the base times the height of the triangle and divide by two.  So the first triangle would have an area of 10, and the second one would have an area of 8. 8 is not equal to 10, so the two given triangles do not have the same area.  So as a result, the last triangle should either have an area of 8 or 10. Now let's look at each triangle and disregard the ones that don't fit the criteria.  The first two are scalene, meaning that their side lengths are all different, so they cannot be the third triangle.  It cannot be the last one either, because that triangle is right, and we know that this triangle cannot be right.  The remaining two are both isosceles and not right, so let's take a look at their areas. The first one has base and height 4 and 6, and the second one has 4 and 4.  So the area of the first option would be 12, and the area of the second option would be 8. Again, base times height over 2 for these.  We want a triangle with an area of either 8 or 10, so the triangle with the area of 8 must be the third triangle, since it fits all of the criteria.  It's isosceles, it's not right, and it has an area of either 8 or 10. So the question asked us, which could be the third one? The answer is this one, letter D.

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