Quick Review Flashcards - Click to flip and test your knowledge!
Question
Which Indian mathematician developed a relationship between the squares of the sides of a right-angled triangle around 600 B.C.?
Answer
The Indian mathematician Buddhayan developed this relationship.
Question
According to the Pythagoras Theorem, what is the relationship between the hypotenuse and the remaining two sides of a right-angled triangle?
Answer
The square on the hypotenuse is equal to the sum of the squares on the remaining two sides.
Question
In the area-based proof of Pythagoras Theorem, what is the geometric relationship between $\triangle GAC$ and square $ABFG$?
Answer
The area of $\triangle GAC$ is half the area of square $ABFG$.
Question
In the area-based proof, why are $\triangle GAC$ and $\triangle BAE$ considered congruent?
Answer
They are congruent by the Side-Angle-Side (SAS) postulate.
Question
In the area-based proof, the area of square $ABFG$ is proved to be equal to the area of which rectangle?
Answer
It is equal to the area of rectangle $AMNE$.
Question
What is the Converse of Pythagoras Theorem?
Answer
If the square on the longest side of a triangle equals the sum of the squares on the other two sides, the angle opposite the longest side is a right-angle.
Question
What is the similarity-based construction used to prove $AC^2 = AB^2 + BC^2$ in $\triangle ABC$ where $\angle ABC = 90^{\circ}$?
Answer
The construction involves drawing a perpendicular $BD$ from the right-angle vertex to the hypotenuse $AC$.
Question
Using similar triangles in the proof of Pythagoras Theorem, what does $BC^2$ equal in terms of the hypotenuse $AC$ and segment $DC$?
Answer
$BC^2$ equals $AC \times DC$.
Question
Using similar triangles in the proof of Pythagoras Theorem, what does $AB^2$ equal in terms of the hypotenuse $AC$ and segment $AD$?
Answer
$AB^2$ equals $AC \times AD$.
Question
Which postulate is used to prove that $\triangle ABC \sim \triangle BDC$ in the similarity-based proof of Pythagoras Theorem?
Answer
The Angle-Angle (A.A.) postulate is used.
Question
In any right-angled triangle, which side is always the largest?
Answer
The hypotenuse is always the largest side.
Question
If $AB$ is the largest side of $\triangle ABC$ and $AB^2 > AC^2 + BC^2$, what type of triangle is it?
Answer
It is an obtuse-angled triangle.
Question
If $AB$ is the largest side of $\triangle ABC$ and $AB^2 < AC^2 + BC^2$, what type of triangle is it?
Answer
It is an acute-angled triangle.
Question
Define 'Pythagorean triplets' using positive numbers $a$, $b$, and $c$ (where $c$ is the largest).
Answer
They are three positive numbers such that $a^2 + b^2 = c^2$.
Question
Provide a common example of a Pythagorean triplet mentioned in the text.
Answer
The numbers 3, 4, and 5 form a Pythagorean triplet.
Question
What is the formula for the width of a street $PQ$ when a ladder of length $L$ reaches heights $h_1$ and $h_2$ on opposite sides?
Answer
$PQ = \sqrt{L^2 - h_1^2} + \sqrt{L^2 - h_2^2}$.
Question
In a right-angled triangle with sides $a$ and $b$, hypotenuse $c$, and altitude $p$ to the hypotenuse, what is the relationship between $p$, $a$, and $b$?
Answer
The relationship is $\frac{1}{p^2} = \frac{1}{a^2} + \frac{1}{b^2}$.
Question
In a triangle where $AD \perp BC$ produced, what is the identity for the side $c$ opposite the obtuse angle?
Answer
The identity is $c^2 = a^2 + b^2 + 2ax$.
Question
In a triangle where $AD \perp BC$ and the angle at $C$ is acute, what is the identity for side $c$ (where $CD = x$)?
Answer
The identity is $c^2 = a^2 + b^2 - 2ax$.
Question
What is the geometric property regarding the sum of the squares of the diagonals of a parallelogram?
Answer
The sum of the squares on the diagonals is equal to the sum of the squares on its sides.
Question
Express the relationship between the side $s$ and diagonals $d_1, d_2$ of a rhombus using Pythagoras Theorem.
Answer
The relationship is $4s^2 = d_1^2 + d_2^2$.
Question
If the lengths of the sides of a triangle are in the ratio $5:12:13$, what specific type of triangle is it?
Answer
It is a scalene right-angled triangle.
Question
In a right-angled triangle, if the hypotenuse is 10 cm and the ratio of the other two sides is $3:4$, what are the lengths of those sides?
Answer
The sides are 6 cm and 8 cm.
Question
What is the area of an isosceles triangle where the equal sides $AB = AC = 12$ cm and the base $BC = 8$ cm?
Answer
The area is $16\sqrt{2}$ $\text{cm}^2$.
Question
In a rhombus with diagonals of 30 cm and 40 cm, what is its perimeter?
Answer
The perimeter is 100 cm.
Question
If a man goes 40 m due north and then 50 m due west, what is his distance from the starting point?
Answer
The distance is $\sqrt{4100}$ m (or $10\sqrt{41}$ m).
Question
In an equilateral triangle with side $x$, what is the length of the altitude $AD$ in terms of $x$?
Answer
The length is $\frac{\sqrt{3}}{2}x$.
Question
In $\triangle ABC$ right-angled at $B$, if $M$ is a point on $BC$, what is the relationship between $AM^2, BC^2, AC^2,$ and $BM^2$?
Answer
The relationship is $AM^2 + BC^2 = AC^2 + BM^2$.
Question
For any point $O$ inside a rectangle $ABCD$, what is the relationship between the distances to the vertices?
Answer
$OB^2 + OD^2 = OC^2 + OA^2$.
Question
The sides of a rectangle are 12 cm and 16 cm; what is the length of its diagonal?
Answer
The length of the diagonal is 20 cm.
Question
If a triangle has sides in the ratio $1 : \sqrt{2} : 1$, why is it a right-angled triangle?
Answer
Because $1^2 + 1^2 = (\sqrt{2})^2$, satisfying the Pythagoras Theorem.
Question
A ladder 13 m long rests against a vertical wall with its foot 5 m from the wall; how high does it reach?
Answer
It reaches a height of 12 m.
Question
If $AD$ is the altitude of an equilateral triangle $ABC$, what is the value of $3AB^2$ in terms of $AD^2$?
Answer
$3AB^2 = 4AD^2$.
Question
In a right-angled triangle $PQR$ at $Q$, with $M$ on $QR$ and $N$ on $PQ$, what does $PM^2 + RN^2$ equal?
Answer
It equals $PR^2 + MN^2$.
Question
In $\triangle ABC$, if $AB > AC$ and $E$ is the mid-point of $BC$ with $AD \perp BC$, what does $AB^2 - AC^2$ equal?
Answer
It equals $2BC \times ED$.
Question
What is the diagonal length of a square with side $x$?
Answer
The diagonal length is $x\sqrt{2}$.
Question
In a quadrilateral $ABCD$, if $\angle A + \angle D = 90^{\circ}$, what is the relationship between its diagonals and sides?
Answer
$AC^2 + BD^2 = AD^2 + BC^2$.
Question
If the sides containing the right angle of a triangle are 4 cm and $4\sqrt{3}$ cm, what is the length of the longest side?
Answer
The longest side is 8 cm.
Question
What is the perimeter of a right-angled triangle with sides 4 cm and $4\sqrt{3}$ cm?
Answer
The perimeter is $(12 + 4\sqrt{3})$ cm.
Question
In a triangle where $AB = AC$ (isosceles) and $AD \perp BC$, what is the property of point $D$?
Answer
Point $D$ is the mid-point of $BC$.
Question
If $BC^2 = AB^2 + AC^2$ in $\triangle ABC$, which angle is $90^{\circ}$?
Answer
$\angle A$ (or $\angle BAC$) is $90^{\circ}$.
Question
Two poles of heights 6 m and 11 m stand 12 m apart; what is the distance between their tips?
Answer
The distance between their tips is 13 m.
Question
In $\triangle ABC$ where $AB=8, BC=6, AC=3$, why is it not a right-angled triangle?
Answer
Because $3^2 + 6^2 \neq 8^2$ ($9 + 36 = 45 \neq 64$).
Question
How is the area of a triangle related to a rectangle if they share the same base and are between the same parallels?
Answer
The area of the triangle is half the area of the rectangle.
Question
In the identity $c^2 = a^2 + b^2 + 2ax$, what does the variable '$x$' represent?
Answer
It represents the length of the projection of one side onto the extension of the base.
Question
If the ratio of the sides of a triangle is $3:4:5$, what is the measure of the angle opposite the side with ratio 5?
Answer
The measure of the angle is $90^{\circ}$.
Question
Statement: In a right-angled triangle, the area of the square on the hypotenuse is equal to the sum of the areas of the squares on the other two sides. Is this true?
Answer
Yes, this is the geometric interpretation of the Pythagoras Theorem.
Question
In $\triangle ABC$, if $AD$ is drawn perpendicular to $BC$ and $AD^2 = BD \times DC$, what is $\angle BAC$?
Answer
$\angle BAC$ is $90^{\circ}$.
Question
What is the length of the diagonal of a rectangle with sides '$l$' and '$b$'?
Answer
The length is $\sqrt{l^2 + b^2}$.
Question
In an isosceles right-angled triangle with hypotenuse $h$ and equal sides $s$, what is the formula for $h^2$?
Answer
$h^2 = 2s^2$.