A(2, 3, k),B(-1, k, -1),and C(4, -3, 2) are t | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Given that $AB = AC$. Using the distance formula,$AB^2 = AC^2$. $AB^2 = (2 - (-1))^2 + (3 - k)^2 + (k - (-1))^2 = 3^2 + (3 - k)^2 + (k + 1)^2 = 9

Vedclass · Accepted Answer

a right-angled isosceles triangle

Vedclass · Answer

(B) Given that $AB = AC$. Using the distance formula,$AB^2 = AC^2$. $AB^2 = (2 - (-1))^2 + (3 - k)^2 + (k - (-1))^2 = 3^2 + (3 - k)^2 + (k + 1)^2 = 9 + 9 - 6k + k^2 + k^2 + 2k + 1 = 2k^2 - 4k + 19$. $AC^2 = (2 - 4)^2 + (3 - (-3))^2 + (k - 2)^2 = (-2)^2 + 6^2 + (k - 2)^2 = 4 + 36 + k^2 - 4k + 4 = k^2 - 4k + 44$. Equating $AB^2 = AC^2$: $2k^2 - 4k + 19 = k^2 - 4k + 44$. $k^2 = 25$. Since $k > 0$,we have $k = 5$. Now,calculate the side lengths: $AB^2 = 2(5)^2 - 4(5) + 19 = 50 - 20 + 19 = 49 \Rightarrow AB = 7$. $AC^2 = 49 \Rightarrow AC = 7$. $BC^2 = (-1 - 4)^2 + (5 - (-3))^2 + (-1 - 2)^2 = (-5)^2 + 8^2 + (-3)^2 = 25 + 64 + 9 = 98$. Since $AB^2 + AC^2 = 49 + 49 = 98 = BC^2$,the triangle satisfies the Pythagorean theorem. Thus,$	riangle ABC$ is a right-angled isosceles triangle.

$A(2, 3, k)$,$B(-1, k, -1)$,and $C(4, -3, 2)$ are the vertices of $\triangle ABC$. If $AB = AC$ and $k > 0$,then $\triangle ABC$ is:

Similar Questions

The triangle formed by the points $(0, 7, 10), (-1, 6, 6), (-4, 9, 6)$ is

If $A(-4, 5, p)$,$B(3, 1, 4)$,and $C(-2, 0, q)$ are the vertices of a triangle $ABC$ and $G(r, q, 1)$ is its centroid,then the value of $2p + q - r$ is equal to

Let $A(4, -2)$,$B(1, 1)$,and $C(9, -3)$ be the vertices of a triangle $ABC$. Then the maximum area of the parallelogram $AFDE$,formed with vertices $D, E$,and $F$ on the sides $BC, CA$,and $AB$ of the triangle $ABC$ respectively,is $\qquad$ .

If $G(2, -1, 2)$ is the centroid of tetrahedron $OABC$ where $O=(0, 0, 0)$ and $G_1$ is the centroid of $\triangle ABC$,then $\left|\overline{O G_1}\right|=$

If the points $(-1, -1, 2), (2, m, 5)$ and $(3, 11, 6)$ are collinear,find the value of $m$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module