List of top Mathematics Questions on Triangles

Let ABC be an equilateral triangle. A new triangle is formed by joining the middle points of all sides of the triangle ABC and the same process is repeated infinitely many times. If P is the sum of perimeters and Q is be the sum of areas of all the triangles formed in this process, then:

In a triangle $ABC$, $BC = 7$, $AC = 8$, $AB = \alpha \in \mathbb{N}$ and $\cos A = \frac{2}{3}$. If \[ 49 \cos(3C) + 42 = \frac{m}{n}, \] where $\gcd(m, n) = 1$, then $m + n$ is equal to ________.

Let $P(\alpha, \beta, \gamma)$ be the image of the point $Q(3, -3, 1)$ in the line \[\frac{x - 0}{1} = \frac{y - 3}{1} = \frac{z - 1}{-1}\]and $R$ be the point $(2, 5, -1)$. If the area of the triangle $PQR$ is $\lambda$ and $\lambda^2 = 14K$, then $K$ is equal to:

If P(6, 1) be the orthocentre of the triangle whose vertices are A(5, –2), B(8, 3) and C(h, k), then the point C lies on the circle.

Consider a triangle \( \triangle ABC \) having the vertices \( A(1, 2) \), \( B(\alpha, \beta) \), and \( C(\gamma, \delta) \) and angles \( \angle ABC = \frac{\pi}{6} \) and \( \angle BAC = \frac{2\pi}{3} \). If the points \( B \) and \( C \) lie on the line \( y = x + 4 \), then \( \alpha^2 + \gamma^2 \) is equal to \( \dots \).

If the orthocentre of the triangle formed by the lines 2x + 3y – 1 = 0, x + 2y – 1 = 0 and ax + by – 1 = 0, is the centroid of another triangle, whose circumecentre and orthocentre respectively are (3, 4) and (–6, –8), then the value of |a– b| is_____.

An equilateral triangle of side \( 4\sqrt{3} \) cm formed out of a sheet is converted into a rectangle such that there is no loss of the area of the triangle. Then the least perimeter of the rectangle (in cm) will be:

If the centroid of a triangle formed by the points \((a,b), (b, c)\) and \((c,a)\) is at the origin, then \(a^3+b^3+c^3 =\)

In \(△ABC\), \(DE || BC\), \(\frac {AD}{DB}=\frac 35\) and \(AC=5.6\) cm, then \(AE =\)\(?\)

In a certain code, COMPUTER is coded as ETUPMOCR. How is INACTIVE written in the same code?

Let C(α, β) be the circumcenter of the triangle formed by the lines
4x+3y=69,
4y-3x=17, and
x+7y=61.
Then (α-β)²+α+β is equal to

Let the ellipse \(E : x^2 + 9y^2 = 9\) intersect the positive x- and y-axes at the points A and B respectively Let the major axis of E be a diameter of the circle C. Let the line passing through A and B meet the circle C at the point P. If the area of the triangle which vertices A, P and the origin O is m/n , where m and n are coprime, then m – n is equal to 

In △ABC, BD is the internal bisector of ∠B meeting AC at D. If CD = 7 cm and AC = 10.5 cm, then AB : BC is

In a \(\angle\)\(ABC\) right angled at A, if \(\triangle\)\(ABC=60°\) and\( AC=4\) units, then length of \(BC\) (in units) is :

If AM and PN are the altitudes of two similar triangles \(∆ABC\) and \(∆PQR\) respectively and \((AB)^ 2 : (PQ) ^2 = 4 : 9\), then \(AM : PN =\)

In the given \(∆ABC\), if \(DE || BC, AE = a\) units, \(EC = b\) units, \(DE = x \) units and \(BC = y\) units, then which of the following is true?

If the centroid of the triangle formed by the points \((3,-5),(-7,4)\) and \((10,-k )\) is at the point \((k, –1)\), then the value of k is

The points A(-5,0), B(5,0) and C(0,4) are the vertices of which triangle?

In the given figure, \(PQ||BC\). If AP=3 cm, BP=2 cm and CQ=3 cm, then AQ= 

\(\triangle ABC \sim \triangle DEF\) and their areas are respectively \(81 \text{ cm}^2 \quad \text{and} \quad 225 \text{ cm}^2\); if \(EF = 5 cm\), then \(BC=\)

In any \(△ABC\), the value of \(\sin \left( \frac{A + B}{2} \right)\) is

In the given figure, \(∠ADE = ∠CBA\), if AD=3.8 cm, AE=3.6 cm, BE= 2.1 cm and BC= 4.2 cm, then DE =

If the ratio of corresponding sides of two similar triangles is 4: 9, then the ratio of areas of these triangles is

In the given figure, AB=AD. ∠ACB=95°+ ∠BAC and ∠BAD=150°. Find ∠ACB.

Let A(3, 0, - 1) , B(2, 10, 6) and C(1, 2, 1) be the vertices of a triangle and M be the mid-point of AC. If G divides BM in the ratio 2:1, then cos( ∠GOA ) (O being the origin) is equal to______.