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The points {A}(0,0), {B}(0,4a -5) and
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08 Jun 2026, 09:24
08 Jun 2026, 09:24
Expert Reply
00:00
Question Stats:
60% (02:32) correct
40% (01:24) wrong based on 5 sessions
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The points \(\mathrm{A}(0,0), \mathrm{B}(0,4 \mathrm{a}-5)\) and \(\mathrm{C}(2 \mathrm{a}+1,2 \mathrm{a}+6)\) form a triangle. If \(\angle A B C=90^{\circ}\) , what is the area of triangle A B C ?
\(\begin{aligned}
& A. 102 \\
& B. 120 \\
& C. 132 \\
& D. 144 \\
& E. 156
\end{aligned}\)
\(\begin{aligned}
& A. 102 \\
& B. 120 \\
& C. 132 \\
& D. 144 \\
& E. 156
\end{aligned}\)
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Re: The points {A}(0,0), {B}(0,4a -5) and
[#permalink]
22 Jun 2026, 04:50
22 Jun 2026, 04:50
1
Expert Reply
Explanation
Given:
\(A(0,0)\)
\(B(0, 4a−5)\)
\(C(2a+1, 2a+6)\)
and \(∠ABC=90.\)
Since \(∠ABC\) is a right angle, vectors BA and BC are perpendicular.
\(BA=(0−(0), 0−(4a−5))=(0, 5−4a)\)
\(BC=((2a+1)−0, (2a+6)−(4a−5))=(2a+1, 11−2a)\)
For perpendicular vectors, their dot product is zero:
\(BA*BC=0\)
\(0(2a+1)+(5−4a)(11−2a)=0\)
\((5−4a)(11−2a)=0\)
Thus, \(a=\frac{5}{4}\) or \(a=\frac{11}{2}\)
Case 1: \(a=\frac{5}{4}\)
\(B=(0,0)\) which coincides with \(A=(0,0),\) so no triangle is formed. Reject this value.
Case 2: \(a=\frac{11}{2}\)
Coordinates become:
\(A=(0,0), B=(0,17), C=(12,17)\)
So the triangle is right-angled at B.
\(AB=17, BC=12\)
Area:\(\frac{1}{2}(AB)(BC)=\frac{1}{2}(17)(12)=102\)
Answer: A
Given:
\(A(0,0)\)
\(B(0, 4a−5)\)
\(C(2a+1, 2a+6)\)
and \(∠ABC=90.\)
Since \(∠ABC\) is a right angle, vectors BA and BC are perpendicular.
\(BA=(0−(0), 0−(4a−5))=(0, 5−4a)\)
\(BC=((2a+1)−0, (2a+6)−(4a−5))=(2a+1, 11−2a)\)
For perpendicular vectors, their dot product is zero:
\(BA*BC=0\)
\(0(2a+1)+(5−4a)(11−2a)=0\)
\((5−4a)(11−2a)=0\)
Thus, \(a=\frac{5}{4}\) or \(a=\frac{11}{2}\)
Case 1: \(a=\frac{5}{4}\)
\(B=(0,0)\) which coincides with \(A=(0,0),\) so no triangle is formed. Reject this value.
Case 2: \(a=\frac{11}{2}\)
Coordinates become:
\(A=(0,0), B=(0,17), C=(12,17)\)
So the triangle is right-angled at B.
\(AB=17, BC=12\)
Area:\(\frac{1}{2}(AB)(BC)=\frac{1}{2}(17)(12)=102\)
Answer: A
gmatclubot
Re: The points {A}(0,0), {B}(0,4a -5) and [#permalink]
22 Jun 2026, 04:50
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