Construct a triangle with , , . Construct an altitude from to .

Understand the Question To construct , we use the given side lengths , , and the included angle . Since is an obtuse angle (), the altitude from vertex to the opposite side lies outside the triangle, requiring the base to be extended past . Step 1 · Construct Triangle 1. Draw a line segment . 2. At point , construct an angle of with . 3. Along this ray, mark a point such that . 4. Join point to point to complete . Step 2 · Construct Altitude from to 1. Extend the line segment past point . 2. With as centre, draw an arc intersecting the extended line at two points, and . 3. With and as centres and a radius greater than , draw intersecting arcs below the line at point . 4. Join to , meeting the extended line at point . is the required altitude perpendicular to extended line . Answer > Triangle and the altitude are constructed as required. Common Mistakes Altitude of an Obtuse Triangle: In an obtuse triangle, altitudes to the sides enclosing the obtuse angle lie outside the triangle. Ensure you extend the base segment past vertex to drop the perpendicular. Protractor Reading: Ensure the angle is measured correctly as an obtuse angle, rather than inadvertently reading from the wrong scale.

A Tale of Three Intersecting Lines | FIO

Question 17

Construct a triangle $TRY$ with $RY = 4\text{ cm}$ , $TR = 7\text{ cm}$ , $\angle R = 140^\circ$ . Construct an altitude from $T$ to $RY$ .

Solution

Understand the Question

To construct $\Delta TRY$ , we use the given side lengths $RY = 4\text{ cm}$ , $TR = 7\text{ cm}$ , and the included angle $\angle R = 140^\circ$ .
Since $\angle R$ is an obtuse angle ( $140^\circ > 90^\circ$ ), the altitude from vertex $T$ to the opposite side lies outside the triangle, requiring the base $RY$ to be extended past $R$ .

Step 1 · Construct Triangle $TRY$

Draw a line segment $RY = 4\text{ cm}$ .
At point $R$ , construct an angle of $140^\circ$ with $RY$ .
Along this ray, mark a point $T$ such that $TR = 7\text{ cm}$ .
Join point $T$ to point $Y$ to complete $\Delta TRY$ .

Step 2 · Construct Altitude from $T$ to $RY$

Extend the line segment $RY$ past point $R$ .
With $T$ as centre, draw an arc intersecting the extended line $RY$ at two points, $A$ and $B$ .
With $A$ and $B$ as centres and a radius greater than $\dfrac{1}{2}AB$ , draw intersecting arcs below the line at point $P$ .
Join $T$ to $P$ , meeting the extended line $RY$ at point $M$ .

$TM$ is the required altitude perpendicular to extended line $RY$ . Diagram 2

Answer

Triangle $TRY$ and the altitude $TM \perp RY$ are constructed as required.

Common Mistakes

Altitude of an Obtuse Triangle: In an obtuse triangle, altitudes to the sides enclosing the obtuse angle lie outside the triangle. Ensure you extend the base segment $RY$ past vertex $R$ to drop the perpendicular.
Protractor Reading: Ensure the $140^\circ$ angle is measured correctly as an obtuse angle, rather than inadvertently reading $40^\circ$ from the wrong scale.

More questions in FIO

Use the points on the circle and/or the centre to form isosceles triangles.

Use the points on the circles and/or their centres to form isosceles and equilateral triangles. The circles are of the same size.

We checked by construction that there are no triangles having sidelengths $3\text{ cm}$ , $4\text{ cm}$ and $8\text{ cm}$ ; and $2\text{ cm}$ , $3\text{ cm}$ and $6\text{ cm}$ . Check if you could have found this without trying to construct the triangle.

Can we say anything about the existence of a triangle for each of the following sets of lengths?

(a) $10\text{ km}$ , $10\text{ km}$ and $25\text{ km}$

(b) $5\text{ mm}$ , $10\text{ mm}$ and $20\text{ mm}$

Which of the following lengths can be the sidelengths of a triangle? Explain your answers. Note that for each set, the three lengths have the same unit of measure.

(a) $2, 2, 5$ (b) $3, 4, 6$ (c) $2, 4, 8$ (d) $5, 5, 8$ (e) $10, 20, 25$ (f) $10, 20, 35$ (g) $24, 26, 28$

Check if a triangle exists for each of the following set of lengths:

(a) 1, 100, 100

(b) 3, 6, 9

(d) 5, 10, 12

Does there exist an equilateral triangle with sides 50, 50, 50? In general, does there exist an equilateral triangle of any sidelength? Justify your answer.

For each of the following, give at least 5 possible values for the third length so there exists a triangle having these as sidelengths (decimal values could also be chosen):

(a) $1$ , $100$

(b) $5$ , $5$

See if you can describe all possible lengths of the third side in each case, so that a triangle exists with those sidelengths. For example, in case (a), all numbers strictly between $99$ and $101$ would be possible.

Construct triangles for the following measurements where the angle is included between the sides:

(a) $3\text{ cm}$ , $75^\circ$ , $7\text{ cm}$

(b) $6\text{ cm}$ , $25^\circ$ , $3\text{ cm}$

Q10

Construct triangles for the following measurements:

(a) $75^\circ$ , $5\text{ cm}$ , $75^\circ$

(b) $25^\circ$ , $3\text{ cm}$ , $60^\circ$

Q11

For each of the following angles, find another angle for which a triangle is (a) possible, (b) not possible. Find at least two different angles for each category:

(a) $30^\circ$

(b) $70^\circ$

(d) $144^\circ$

Q12

Determine which of the following pairs can be the angles of a triangle and which cannot:

(a) $35^\circ$ , $150^\circ$

(b) $70^\circ$ , $30^\circ$

(d) $50^\circ$ , $150^\circ$

Q13

Find the third angle of a triangle (using a parallel line) when two of the angles are:

(a) $36^\circ, 72^\circ$

(b) $150^\circ, 15^\circ$

(d) $75^\circ, 45^\circ$

Q14

Can you construct a triangle all of whose angles are equal to $70^\circ$ ? If two of the angles are $70^\circ$ what would the third angle be? If all the angles in a triangle have to be equal, then what must its measure be? Explore and find out.

Q15

Here is a triangle in which we know $\angle B = \angle C$ and $\angle A = 50^\circ$ . Can you find $\angle B$ and $\angle C$ ?

Q16

Construct a triangle $ABC$ with $BC = 5\text{ cm}$ , $AB = 6\text{ cm}$ , $CA = 5\text{ cm}$ . Construct an altitude from $A$ to $BC$ .

Q17

Construct a triangle $TRY$ with $RY = 4\text{ cm}$ , $TR = 7\text{ cm}$ , $\angle R = 140^\circ$ . Construct an altitude from $T$ to $RY$ .

Q18

Construct a right-angled triangle $\Delta \text{ABC}$ with $\angle B = 90^\circ$ , $\text{AC} = 5 \text{ cm}$ . How many different triangles exist with these measurements?

[Hint: Note that the other measurements can take any values. Take $\text{AC}$ as the base. What values can $\angle A$ and $\angle C$ take so that the other angle is $90^\circ$ ?]

Q19

Through construction, explore if it is possible to construct an equilateral triangle that is: (i) right-angled (ii) obtuse-angled.

Also construct an isosceles triangle that is: (i) right-angled (ii) obtuse-angled.