Question

do the points (3, 2) (-2 -3) and (2 3) form a triangle? if so name the type of triangle formed




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Answer 1

Answer:

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Step-by-step explanation:

Hence, we are given with the points A (3, 2), B (-2, -3) and C (2, 3)

Let us consider the figure:

Now, we have to check whether these vertices are the vertices of a triangle or not.

For that first we have to use the distance formula. Let P(x1,y1) and Q(x2,y2) be two points then the distance between P and Q is given by the distance formula:

PQ=(x1−y1)2+(x2−y2)2−−−−−−−−−−−−−−−−−−√

By applying the distance formula we have to find the values of AB, BC and AC.

First we can find AB. We have A (3, 2) and B (-2 -3) where (x1,y1)=(3,2) and (x2,y2)=(−2,3). Hence, we will get:

AB=(−2−3)2+(−3−2)2−−−−−−−−−−−−−−−−−−√AB=(−5)2+(−5)2−−−−−−−−−−−√AB=25+25−−−−−−√AB=50−−√

Now, we can factorise 50, we have:

50=2×5×5

Now, AB can be written as:

AB=2×5×5−−−−−−−√AB=2×52−−−−−√AB=2–√×52−−√AB=2–√×5AB=52–√

Hence, we got the value of AB=52–√.

Now, we have to find the value of BC, B (-2, -3) and C (2, 3) where (x1,y1)=(−2,−3) and (x2,y2)=(2,3).Hence we obtain:

BC=(2−(−2))2+(3−(−3))2−−−−−−−−−−−−−−−−−−−−√BC=(2+2)2+(3+3)2−−−−−−−−−−−−−−−√BC=42+62−−−−−−√BC=16+36−−−−−−√BC=52−−√

Now, we can factorise 52, we will get:

52=13×2×2

Now, AB can be written as:

BC=13×2×2−−−−−−−−√BC=13×22−−−−−−√BC=13−−√×22−−√BC=13−−√×2BC=213−−√

Next, we have to find the value of AC. We have A (3, 2) and C (2, 3) where (x1,y1)=(3,2) and (x2,y2)=(2,3).Hence we obtain:

AC=(3−2)2+(2−3)2−−−−−−−−−−−−−−−√AC=12+(−1)2−−−−−−−−−√AC=1+1−−−−√AC=2–√

Now let us take the square of AB, BC and AC, we will get:

AB=50−−√AB2=(50−−√)2AB2=50

Similarly, we will obtain:

BC=52−−√BC2=(52−−√)2BC2=52

Next, we will get:

AC=2–√AC2=(2–√)2AC2=2

Now, from the above data we can write:

52=50+2

Hence, we can write:

BC2=AB2+AC2

The above equation satisfies the Pythagoras theorem which says that in a right angled triangle the square of the hypotenuse is the sum of the squares of its base and altitude. Here, BC is the hypotenuse, AB is the altitude and AC is the base of the triangle ΔABC.

We know that Pythagoras theorem is only applicable for a right angled triangle. Therefore, by Pythagoras theorem, we can say that the triangle ΔABC is a right angled triangle.

Hence, we can say that the points A (3, 2), B (-2, -3) and C (2, 3) form the vertices of a right angled triangle.