C-03 — Geometry

The renowned Greek scholar who described an extremely important theorem about right-angled triangles is —

The lifespan of Pythagoras is —

The theorem of Pythagoras was known to —

About how many years before Pythagoras was the theorem already known?

In a right-angled triangle the area of the square drawn on the hypotenuse is equal to —

In right $\triangle ABC$ with $\angle BAC = 90°$, the hypotenuse is —

If two sides adjacent to the right angle of a right-angled triangle are $b = 8$ cm and $c = 6$ cm, then the length of the hypotenuse is —

In right $\triangle ABC$ with $\angle A = 90°$, $BC^2 =$

If the hypotenuse of a right-angled triangle is 13 cm and one side is 5 cm, the other side is —

According to the converse of the theorem of Pythagoras, if in $\triangle ABC$, $a^2 = b^2 + c^2$, then the angle opposite to side $a$ is —

In $\triangle ABC$, $AB = 6$ cm, $BC = 10$ cm, $AC = 8$ cm. Which angle is a right angle?

Consider the following statements:

The orthogonal projection of a point on a definite straight line is —

The orthogonal projection of any point on a definite straight line is —

To find the orthogonal projection of a line segment $AB$ on a line $XY$, perpendiculars are drawn from —

If a line segment $AB$ is parallel to the line $XY$, its orthogonal projection $A'B'$ on $XY$ equals —

The orthogonal projection of a line perpendicular to a given straight line is —

The length of the orthogonal projection of a perpendicular onto a line is —

Consider the following statements about orthogonal projection:

In $\triangle ABC$, $\angle BCA$ is obtuse and $CD$ is the orthogonal projection of $AC$ on the extended $BC$. Then —

In $\triangle ABC$, $\angle ACB$ is acute and $CD$ is the orthogonal projection of $AC$ on $BC$. Then —

Theorems 3 and 4 of this chapter are also known as —

In a right-angled triangle, the orthogonal projection of one perpendicular side on the other perpendicular side is —

Which inequality holds when $\angle ACB$ is obtuse in $\triangle ABC$?

Which inequality holds when $\angle ACB$ is acute in $\triangle ABC$?

In $\triangle ABC$, consider the following:

According to the textbook, the lifespan of Apollonius is given as —

According to the theorem of Apollonius, if $AD$ is the median to side $BC$ of $\triangle ABC$, then —

The theorem of Apollonius states that the sum of the areas of the squares drawn on any two sides of a triangle is equal to —

In $\triangle ABC$ with sides $a, b, c$ opposite to vertices $A, B, C$ and medians $d, e, f$ from $A, B, C$ respectively, $d^2 =$

With the same notation, $e^2 =$

With the same notation, $f^2 =$

In any triangle, the relation between sides and medians is —

In a right-angled triangle with hypotenuse $c$ and medians $d, e, f$, —

In $\triangle ABC$, $\angle B = 60°$. Then $AC^2 =$

In $\triangle ABC$, $\angle B = 120°$. Then $AC^2 =$

In $\triangle ABC$, $\angle C = 90°$ and $D$ is the midpoint of $BC$. Then $AB^2 =$

In $\triangle ABC$, $AD$ is the perpendicular on $BC$ and $BE$ is the perpendicular on $AC$. Then —

In $\triangle ABC$, side $BC$ is trisected at points $P$ and $Q$. Then $AB^2 + AC^2 =$

In $\triangle ABC$, $AB = AC$ and $P$ is a point on $BC$. Then $AB^2 - AP^2 =$

If the three medians of $\triangle ABC$ meet at $G$, then $AB^2 + BC^2 + AC^2 =$

Two polygons having the same number of sides with successive equal angles are called —

Two polygons of the same number of sides are similar if —

Which of the following pairs is equiangular but not similar?

A square $EFGH$ and a rhombus $KLMN$ (which is not a square) are —

If two triangles are equiangular, their corresponding sides are proportional. This is —

According to the corollary of Theorem 6, if two triangles are equiangular, they are —

If two angles of one triangle are equal to two angles of another triangle, the two triangles are —

If the three sides of two triangles are proportional, the angles opposite to corresponding sides are —

If one angle of one triangle is equal to an angle of another triangle and the sides adjoining the equal angles are proportional, the two triangles are —

The ratio of the areas of two similar triangles is equal to the ratio of —

Two similar triangles have corresponding sides in the ratio $3 : 2$. The ratio of their areas is —

The point of intersection of the perpendicular bisectors of the sides of a triangle is the —

The point of intersection of the three medians of a triangle is the —

The centroid of a triangle divides each median in the ratio —

The point of intersection of the perpendiculars drawn from each vertex of a triangle to the opposite side is the —

The distance between the orthocenter and a vertex of a triangle is —

The circumcenter, the centroid and the orthocenter of any triangle are —

The radius of the nine-point circle is —

The center of the nine-point circle is the midpoint of the line segment joining —

How many points lie on the nine-point circle of a triangle?

Which of the following points lie on the nine-point circle of a triangle?

The theorem of Brahmagupta applies to a cyclic quadrilateral with —

According to Brahmagupta's theorem, the perpendicular drawn from the point of intersection of the diagonals to one side of a cyclic quadrilateral —

In a cyclic quadrilateral $ABCD$ with perpendicular diagonals $AC$ and $BD$ meeting at $M$, $ME \perp BC$ and the extended $EM$ meets $AD$ at $F$. Then —

Ptolemy's theorem for a cyclic quadrilateral $ABCD$ with diagonals $AC$, $BD$ states —

According to Ptolemy's theorem, in any cyclic quadrilateral, the rectangle contained by the two diagonals equals —

In $\triangle PQR$ with $\angle PQR = 90°$, $D, E, F$ are the midpoints of $PQ, QR, PR$ respectively. The medians $PE, QF, DR$ meet at a point that is the —

In $\triangle PQR$ with $\angle PQR = 90°$ and $E$ the midpoint of $QR$, then $PR^2 =$

In $\triangle PQR$ with $\angle PQR = 90°$ and $QF \perp PR$ at $F$, then $QF^2 =$

From any point $P$ on the circumcircle of $\triangle ABC$, perpendiculars $PD$ and $PE$ are drawn on $BC$ and $CA$ respectively. The line segment $ED$ intersects $AB$ at $O$. Then $PO$ is —

In right $\triangle ABC$ with $\angle C = 90°$ and $CD$ the perpendicular drawn from $C$ on the hypotenuse $AB$, then $CD^2 =$

In $\triangle ABC$, the perpendiculars $AD, BE, CF$ from the vertices to the opposite sides intersect at $O$. Then —

A semicircle is drawn on diameter $AB$. Two of its chords $AC$ and $BD$ intersect at point $P$. Then $AB^2 =$

In an isosceles $\triangle ABC$ with $AB = AC$, $AD$ is the perpendicular from $A$ to base $BC$. If $R$ is the circumradius of the triangle, then $AB^2 =$

The bisector of $\angle A$ of $\triangle ABC$ meets $BC$ at $D$ and the circumcircle at $E$. Then $AD^2 =$

In $\triangle ABC$, $BE$ and $CF$ are perpendiculars on $AC$ and $AB$ respectively. Then $\dfrac{\triangle ABC}{\triangle AEF} =$

In $\triangle PQR$, the medians $PM, QN, RS$ intersect at $O$. The point $O$ divides $PM$ (from vertex $P$) in the ratio —

In $\triangle PQR$ with $M$ the midpoint of $QR$, $PQ^2 + PR^2 =$

In $\triangle PQR$ with centroid $O$, $PQ^2 + QR^2 + PR^2 =$

Length of each of the three medians of an equilateral triangle is 3 cm. What is the length of each side?

In $\triangle ABC$,

In $\triangle ABC$, $D, E, F$ are midpoints of $BC, AC, AB$ respectively. The medians of the triangle intersect at point $G$. The name of point $G$ is —

With the same figure (medians from midpoints $D, E, F$), which statement is consistent with the theorem of Apollonius applied to $\triangle ABC$?

In $\triangle ABC$, $S$ is the circumcenter, $G$ is the centroid and $O$ is the orthocenter. Then $S, G, O$ are —

In $\triangle ABC$ with $\angle C$ acute, $BC = a$, $AC = b$, $CD$ is the orthogonal projection of $AC$ on $BC$, and $CE$ is the orthogonal projection of $BC$ on $AC$. Then —

If the lengths of the three sides of a triangle are 4 cm, 5 cm and 6 cm, the sum of the squares of the medians of the triangle is —

If the sum of the squares of the medians of a right-angled triangle is 37.5 sq units, the length of the hypotenuse is —

The radius of the circumcircle of an equilateral triangle is 3 cm. The length of each side of the triangle is —