# Topics in Grade 10 Math

## Linear Systems (Fraction and Decimals)

### example question:

Solve. Remember to clear fractions and decimals first. $$\frac{x}{2}+\frac{y}{5}=2$$ $$\frac{3x}{2}-\frac{y}{5}=6$$## Linear Systems (General Word Problems)

### example question:

The sum of two numbers is 250. Their difference is 74. Find the numbers.## Linear Systems (Investment Problems)

### example question:

Karin deposited a total of 10,000 dollars in two separate accounts. One account paid 5% interest per annum, and the other paid 8% interest per annum. If the total interest earned after one year was $620, how much was invested at each rate?## Linear Systems (Mixture Problems)

### example question:

Bess wants to make an almond cashew mix. If the almonds cost \($\)2.50/kg and cashews cost \($\)3.50/kg, then how many kilograms of each does she need to make 50 kilograms of mix that will sell for \($\)2.90/kg?## Linear Systems (Money Problems)

### example question:

A vending machine that only accepts dimes and quarters contains 36 coins totaling \($7.20\). How many of each coin does the machine contain?## Linear Systems DST (Two Part Trip)

### example question:

Adrielle traveled to her cottage 340 km away. Part of the trip was by bus that traveled 50km/h, and the other part of the trip was by car at 80 km/h. The total trip took 5 hours.

a) How many hours did she travel by car?

b) How many hours did she travel by bus?

c) How many kilometres did she travel by car?

d) How many kilometres did she travel by bus?## Linear Systems DST (Current \ Wind)

### example question:

Colin took a boat trip 120 km upstream that lasted 5 hours. The return trip lasted 4 hours.

a) find the speed of the boat in still water.

b) find the speed of the current.## Linear Systems DST (Two Different Times)

### example question:

Kevin left his house driving at 40 km/h. His brother Eric followed in his car 1 hour later traveling at 50 km/h.

a) At what distance from their home did Eric catch up to Kevin?

b) How long had Kevin been driving when they meet up?

c) How long had Eric been driving when they meet up?## Length of a Line Segment (Analytic Geometry)

### example question:

Use the formula below to determine the length between the points (2,1) and (5,7) $$\ell=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$## Midpoint of a Line Segment

### example question:

Use the formula below to find the midpoint between A(3,2) and B(5,8). $$\text{midpoint}=\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$$## Equation of a Circle

### example question:

Find the radius and the center of the circle that is defined by the equation: $$x^2+y^2=25$$ The general equation for a circle is $$x^2+y^2=r^2$$## Slope of a Line Segment

### example question:

Use the formula below to determine the slope between points A(3,2), and B(4,7) $$ m=\frac{y_2-y_1}{x_2-x_1}$$## Equations of Lines (Part 1)

### example question:

Use the formula below to determine the equation of the line, in standard form, with a slope of 2 and passes through the point (1,3). $$ y-y_1=m(x-x_1)$$## x-intercept, y-intercept & Slope (Parallel and Perpendicular)

### example question:

Find the parallel and perpendicular slopes to the line: $$3x+4y-12=0$$## Equations of Lines (Part 2)

### example question:

Determine the equation of the line that is

parallel to 2x-y-5=0 and passes through (2,5).## Equations of Lines (Medians and Centroids)

### example question:

Given a triangle with vertices A(0,9), B(14,1) and C(-2,-1) find:

a) The equation of the median from A to BC.

b) The equation of the median from C to AB.

c) The centroid of the triangle ABC.## Equations of Lines (Altitudes and Orthocentres)

### example question:

Given a triangle with vertices A(5,1), B(0,-9) and C(-9,3) find:

a) The equation of the altitude from A to BC.

b) The equation of the altitude from C to AB.

c) The orthocentre of the triangle ABC.## Equations of Lines (Perpendicular / Right Bisectors and Circumcentre)

### example question:

Given a triangle with vertices A(9,10), B(3,-8) and C(-7,2) find:

a) The equation of the perpendicular bisector of BC.

b) The equation of the perpendicular bisector of AB.

c) The circumcentre of the triangle ABC.## Distance from a Point to a Line

### example question:

Find the shortest distance from the point (3,4) to the line x=-2.## Expanding and Simplifying Polynomial Expressions (Part 1)

### example question:

Expand and Simplify $$2x(x-4)-3x(x-3) $$## Special Products (Product of Sum and Difference)

### example question:

Expand and Simplify $$(x+5)(x-5) $$## Expanding and Simplifying Polynomial Expressions (Part 2)

### example question:

Expand and Simplify $$(2x+3)(x-4)+(x+2)^2 $$## Trinomial Factoring (a$\neq$1) (Magic Box)

### example question:

Factor ( using the magic box ) $$4x^2+17x+4 $$## Graphing Quadratic Functions (Vertex Form)

### example question:

State the following and graph the function, given: $$ y=(x-3)^2-4$$

Direction of opening.

The coordinates of the vertex.

The equation of the axis of symmetry.

The domain and range.

The maximum or minimum value.

How the parabola is stretched or compressed if applicable.

## Quadratic Functions (Word Problems Part 1)

### example question:

The equation shows the height (h) of a baseball in metres as a function of time (t) in seconds. $$ h=-5(t-2)^2+21$$ What was the maximum height?

At what time did the ball reach its maximum height?

What is the height of the ball after 1 second?

What was the initial height of the ball?

At what time does the ball hit the ground?## Quadratic Functions (Complete the Square)

### example question:

Write the function in vertex form $y=a(x-h)^2+k $ and state the coordinates of the vertex and the maximum or minimum value. $$y=x^2+6x+1$$## Quadratic Functions (Word Problems Part 2)

### example question:

The equation shows the height (h) of a baseball in metres as a function of time (t) in seconds. $$ h=-5t^2+40t+1$$ What was the maximum height?

At what time did the ball reach its maximum height?

What is the height of the ball after 1 second?

What was the initial height of the ball?

At what time does the ball hit the ground?

## Quadratic Equations (Word Problems Part 1)

### example question:

The width of a rectangle is 1cm shorter than its length. If the area is \(6cm^2\), what are the dimensions?## Quadratic Equations (Solve using the Quadratic Formula)

### example question:

The quadratic formula is: $$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$ Solve using the quadratic formula. $$ x^2-x-12=0 $$## Quadratic Equations (Word Problems Part 2)

### example question:

The hypotenuse of a right triangle is 25 cm. The sum of the lengths of the other two sides is 30 cm. Find the lengths of the the sides.## Similar Triangles

### example question:

(Introductory trigonometry / trig )

Find the values of $x$ and $y$.## Using Your Calculator

### example question:

(Introductory trigonometry / trig )

Find the following to the nearest thousandth.$$ $$ $\sin 15°=$ $$ $$ $\cos 73°=$## sohcahtoa $(S=\frac{O}{H},\;C=\frac{A}{H},\;T=\frac{O}{A})$

### example question:

Find $x$.

( use trig ratios )

( trigonometry / trig )## Word problems (Part 1) (Trigonometry)

### example question:

The sun is at an angle of elevation of 40°. A tree casts a shadow of 25m. How tall is the tree?

( Use trig ratios )

( trigonometry / trig )## Solving Triangles (Advanced Trigonometry) (Two Triangles)

### example question:

Find $BC$.

( Use trig ratios )

( trigonometry / trig )## Sine Law $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$ or $\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{sin C}{c}$

### example question:

Find $x$.

( trigonometry / trig )## Cosine Law (Unknown side - SAS) $a^2=b^2+c^2-2(b)(c)(\cos A)$

### example question:

Find $x$.

( trigonometry / trig )## Cosine Law (Unknown Angle - SSS) $\cos A=\frac{b^2+c^2-a^2}{2bc}$

### example question:

Find $\angle C$.

( trigonometry / trig )

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